aa r X i v : . [ m a t h - ph ] J un RESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS
FR´ED´ERIC KLOPP
Dedicated to Johannes Sj¨ostrand on the occasion of his seventieth birthday.
Abstract.
The present paper is devoted to the study of resonances for one-dimensional quantumsystems with a potential that is the restriction to some large box of an ergodic potential. For discretemodels both on a half-line and on the whole line, we study the distributions of the resonances inthe limit when the size of the box where the potential does not vanish goes to infinity. For periodicand random potentials, we analyze how the spectral theory of the limit operator influences thedistribution of the resonances.
R´esum´e.
Dans cet article, nous ´etudions les r´esonances d’un syst`eme unidimensionnel plong´e dansun potentiel qui est la restriction `a un grand intervalle d’un potentiel ergodique. Pour des mod`elesdiscrets sur la droite et la demie droite, nous ´etudions la distribution des r´esonances dans la limitede la taille de boˆıte infinie. Pour des potentiels p´eriodiques et al´eatoires, nous analysons l’influencede la th´eorie spectrale de l’op´erateur limite sur la distribution des r´esonances. Introduction
Consider V : Z → R a bounded potential and, on ℓ ( Z ), the Schr¨odinger operator H = − ∆ + V defined by ( Hu )( n ) = u ( n + 1) + u ( n −
1) + V ( n ) u ( n ) , ∀ n ∈ Z , for u ∈ ℓ ( Z ).The potentials V we will deal with are of two types: • V periodic; • V = V ω , the random Anderson model, i.e., the entries of the diagonal matrix V are inde-pendent identically distributed non constant random variable.The spectral theory of such models has been studied extensively (see, e.g., [19]) and it is well knownthat • when V is periodic, the spectrum of H is purely absolutely continuous; • when V = V ω is random, the spectrum of H is almost surely pure point, i.e., the operatoronly has eigenvalues; moreover, the eigenfunctions decay exponentially at infinity.Pick L ∈ N ∗ . The main object of our study is the operator(0.1) H L = − ∆ + V J − L +1 ,L K when L is large. Here, J − L + 1 , L K is the integer interval {− L + 1 , · · · , L } and J a,b K ( n ) = 1 if a ≤ n ≤ b and 0 if not.For L large, the operator H L is a simple Hamiltonian modeling a large sample of periodic or randommaterial in the void. It is well known in this case (see, e.g., [43]) that not only does the spectrum Mathematics Subject Classification.
Key words and phrases.
Resonances; random operators; periodic operators.It is a pleasure to thank N. Filonov for interesting discussions at the early stages of this work and, T.T. Phong,C. Shirley and M. Vogel for pointing out misprints in previous versions of the article.This work was partially supported by the grant ANR-08-BLAN-0261-01. of H L be of importance but also its (quantum) resonances that we will now define.As V J − L +1 ,L K has finite rank, the essential spectrum of H L is the same as that of the discreteLaplace operator, that is, [ − , H L has only discrete eigenvalues associated to exponentially decaying eigen-functions.We are interested in the resonances of the operator H L in the limit when L → + ∞ . They aredefined to be the poles of the meromorphic continuation of the resolvent of H L through ( − , H L (see Theorem 1.1 and, e.g., [43]). The resonances widths, that is,their imaginary part, play an important role in the large time behavior of e − itH L , especially theresonances of smallest width that give the leading order contribution (see [43]). x xx x x −2 2 −2 2 x x xx Figure 1: The meromorphic continuationQuantum resonances are basic objects in quantum theory. They have been the focus of vast num-ber of studies both mathematical and physical (see, e.g., [43] and references therein). Our purposehere is to study the resonances of H L in the asymptotic regime L → + ∞ . As L → + ∞ , H L converges to H in the strong resolvent sense. Thus, it is natural to expect that the differences inthe spectral nature between the cases V periodic and V random should reflect into differences inthe behavior of the resonances in both cases. We shall see below that this is the case. To illustratethis as simply as possible, we begin with stating three theorems, one for periodic potentials, twofor random potentials, that underline these different behaviors. These results can be considered asparadigmatic for our main results presented in section 1.The scattering theory or the closely related questions of resonances for the operator (0.1) or forclosely related one-dimensional models has already been discussed in various works both in themathematical and physical literature (see, e.g., [12, 11, 29, 26, 40, 9, 27, 4, 25, 41]). We will makemore comments on the literature as we will develop our results in section 1.0.1. When V is periodic. Assume that V is p -periodic ( p ∈ N ∗ ) and does not vanish identically.Consider H = − ∆ + V and let Σ Z be its spectrum, ◦ Σ Z be its interior and E N ( E ) be itsintegrated density of states, i.e., the number of states of the system per unit of volume belowenergy E (see section 1.2 and, e.g., [39] for precise definitions and details). Theorem 0.1.
There exist • D , a discrete (possibly empty) set of energies in ( − , ∩ ◦ Σ Z , • a function h that is real analytic in a complex neighborhood of ( − , and that does vanishon ( − , \ D such that, for I ⊂ ( − , \ D , a compact interval such that either I ∩ Σ Z = ∅ or I ⊂ ◦ Σ Z , thereexists c > such that for L sufficiently large s.t. L ∈ p N , one has • if I ∩ Σ Z = ∅ , then H L has no resonance in I + i [ − c , • if I ⊂ ◦ Σ Z , one has – there are plenty of resonances in I + i [ − c , ; more precisely, (0.2) { z ∈ I + i [ − c , , z resonance of H L } L = Z I dN ( E ) + o (1) ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 3 where o (1) → as L → + ∞ ; – let ( z j ) j the resonances of H L in I + i [ − c , ordered by increasing real part; then, (0.3) L · Re ( z j +1 − z j ) ≍ and L · Im z j = h ( Re z j ) + o (1) , the estimates in (0.3) being uniform for all the resonances in I + i [ − c , when L → + ∞ . After rescaling their width by L , resonances are nicely inter-spaced points lying on an analyticcurve (see Fig. 2). We give a more precise description of the resonances in Theorem 1.3 andPropositions 1.1 and 1.2. In particular, we describe the set of energies D and the resonancesnear these energies: they lie further away from the real axis, the maximal distance being of order L − log L (see Fig. 3). Theorem 0.1 only describes the resonances closest to the real axis. Insection 1.2, we also give results on the resonances located deeper into the lower half of the complexplane. Iresonances E resonances Figure 2: The rescaled resonances for the periodic (left part) and the random (right part) potential0.2.
When V is random. Assume now that V = V ω is the Anderson potential, i.e., its entriesare i.i.d. distributed uniformly on [0 ,
1] to fix ideas. Consider H = − ∆ + V ω . Let Σ be itsalmost sure spectrum (see, e.g., [33]), E n ( E ), its density of states (i.e. the derivative of theintegrated density of states, see section 1.2 and, e.g., [33]) and E ρ ( E ), its Lyapunov exponent(see section 1.3 and, e.g., [33]). The Lyapunov exponent is known to be continuous and positive(see, e.g., [5]); the density of states satisfies n ( E ) > E ∈ Σ (see, e.g., [5]).Define H ω,L := − ∆ + V ω J − L +1 ,L K . We prove Theorem 0.2.
Pick I ⊂ ( − , , a compact interval. Then, • if I ∩ Σ = ∅ then, there exists c I > such that, ω -a.s., for L sufficiently large, { z resonance of H ω,L in I + i ( − c I , } = ∅ ; • if I ⊂ ◦ Σ then, for any c > , ω -a.s., one has lim L → + ∞ L (cid:8) z resonance of H ω,L in I + i (cid:0) −∞ , − e − cL (cid:3)(cid:9) = Z I min (cid:18) cρ ( E ) , (cid:19) n ( E ) dE. As the first statement of Theorem 0.2 is clear, let us discuss the second. Define c + := max E ∈ I ρ ( E ).For c ≥ c + , ω -a.s., for L large, the number of resonances in the strip { Re z ∈ I, Im z ≤ − e − cL } is approximately L Z I n ( E ) dE ; thus, in { Re z ∈ I, − e c + L ≤ Im z < } , one finds at most o ( L )resonances. We shall see that, for δ > ω -a.s., for L large, the strip { Re z ∈ I, − e ( c + + δ ) L ≤ Im z < } actually contains no resonance (see Theorem 1.6).Define c − := min E ∈ I ρ ( E ). For c ≤ c − , ω -a.s., for L large, the strip { Re z ∈ I, Im z ≤ − e − cL } contains approximately c L Z I n ( E ) ρ ( E ) dE resonances. We shall see that, for κ ∈ [0 , FR´ED´ERIC KLOPP resonances in the strip { Re z ∈ I, Im z ≤ − e − L κ } is O ( L κ ), thus, o ( L ) (cf. Theorem 1.10).One can also describe the resonances locally. Therefore, fix E ∈ ( − , ∩ ◦ Σ such that n ( E ) > z Ll ( ω )) l be the resonances of H ω,L . We first rescale them. Define(0.4) x Ll ( ω ) = 2 n ( E ) L (Re z Ll ( ω ) − E ) and y Ll ( ω ) = − Lρ ( E ) log | Im z Ll ( ω ) | . Consider now the two-dimensional point process ξ L ( E , ω ) = X z Ll resonances of H ω,L δ ( x Ll ( ω ) ,y Ll ( ω )) . We prove
Theorem 0.3.
The point process ξ L converges weakly to a Poisson process of intensity in R × [0 , . In the random case, the structure of the (properly rescaled) resonances is quite different from thatin the periodic case (see Fig. 2). The real parts of the resonances are scaled in such a way that thattheir average spacing becomes of order one. By Theorem 0.2, the imaginary parts are typicallyexponentially small (in L ); when the resonances are rescaled as in (0.4), their imaginary parts arerewritten on a logarithmic scale so as to become of order 1 too. Once rescaled in this way, the localpicture of the resonances of H ω,L is that of a two-dimensional cloud of Poisson points (see the righthand side of fig. 2).Theorem 0.3 is the analogue for resonances of the well known result on the distribution of eigenvaluesand localization centers for the Anderson model in the localized phase (see, e.g., [31, 17, 13]).As in the case of the periodic potential, Theorem 0.3 only describes the resonances closest to thereal axis. In section 1.3, we also give results on resonances located deeper into the lower half ofthe complex plane. Up to distances of order L −∞ to the real axis, the cloud of resonances (onceproperly rescaled) will have the same Poissonian behavior as described above (see Theorem 1.4).Besides proving Theorems 0.1 and 0.3, the goal of the paper is to describe the statistical propertiesof the resonances and relate them (the distribution of the resonances, the distribution of the widths)to the spectral characteristics of H = − ∆ + V , possibly to the distribution of its eigenvalues (see,e.g., [14]).As they can be analyzed in a very similar way, we will discuss three models: • the model H L defined above, • its analogue on the half-line N , i.e., on H L , we impose an additional Dirichlet boundarycondition at 0, • the “half-infinite” model on ℓ ( Z ), that is,(0.5) H ∞ = − ∆ + W where ( W ( n ) = 0 for n ≥ W ( n ) = V ( n ) for n ≤ − V is chosen as above, periodic or random.Though in the present paper we restrict ourselves to discrete models, it is clear that continuousone-dimensional models can be dealt with essentially using the methods developed in the presentpaper. 1. The main results
We now turn to our main results, a number of which were announced in [23]. Pick V : Z → R abounded potential and, for L ∈ N , consider the following operators: • H Z L = − ∆ + V J ,L K on ℓ ( Z ); ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 5 • H N L = − ∆ + V J ,L K on ℓ ( N ) with Dirichlet boundary conditions at 0; • H ∞ defined in (0.5). Remark 1.1.
Here, with “Dirichlet boundary condition at 0”, we mean that H N L is the operator H Z L restricted to the subspace ℓ ( N ), i.e., if Π : ℓ ( Z ) → ℓ ( N ) is the orthogonal projector on ℓ ( N ),one has H N L = Π H Z L Π. In the literature, this is sometime called “Dirichlet boundary condition at −
1” (see, e.g., [39]).For the sake of simplicity, in the half line case, we only consider Dirichlet boundary conditions at0. But the proofs show that these are not crucial; any self-adjoint boundary condition at 0 woulddo and, mutandi mutandis, the results would be the same.Note also that, by a shift of the potential V , replacing L by L + L ′ , we see that studying H Z L isequivalent to studying H L,L ′ = − ∆ + V J − L ′ ,L K on ℓ ( Z ). Thus, to derive the results of section 0from those in the present section, it suffices to consider the models above, in particular, H Z L .For the models H N L and H Z L , we start with a discussion of the existence of a meromorphic continu-ation of the resolvent, then, study the resonances when V is periodic and finally turn to the casewhen V is random.As H ∞ is not a relatively compact perturbation of the Laplacian, the existence of a meromorphiccontinuation of its resolvent depends on the nature of V ; so, it will be discussed when specializingto V periodic or random. Remark 1.2 (Notations) . In the sequel, we write a . b if for some C > a or b ), one has a ≤ Cb . We write a ≍ b if a . b and b . a .1.1. The meromorphic continuation of the resolvent.
One proves the well known and simple
Theorem 1.1.
The operator valued functions z ∈ C + ( z − H N L ) − and z ∈ C + ( z − H Z L ) − )admit a meromorphic continuation from C + to C \ (( −∞ , − ∪ [2 , + ∞ )) through ( − , (see Fig. 1)with values in the operators from l comp to l loc .Moreover, the number of poles of each of these meromorphic continuations in the lower half-planeis at most equal to L . The resonances are defined to be the poles of this meromorphic continuation (see Fig. 1).1.2.
The periodic case.
We assume that, for some p >
0, one has(1.1) V n + p = V n for all n ≥ . Let Σ N be the spectrum of H N = − ∆ + V acting on ℓ ( N ) with Dirichlet boundary condition at 0and Σ Z be the spectrum of H Z = − ∆ + V acting on ℓ ( Z ). One has the following description forthese spectra: • Σ Z is a union of intervals, i.e., Σ Z := σ ( H ) = p [ j =1 [ E − j , E + j ] where E − j < E + j (1 ≤ j ≤ p )and a + j − ≤ E − j (2 ≤ j ≤ p ) (see , e.g., [42]); the spectrum of H Z is purely absolutelycontinuous and the spectral resolution can be obtained via a Bloch-Floquet decomposition(see, e.g., [42]); • on ℓ ( N ) (see, e.g., [34]), one has – Σ N = Σ Z ∪ { v j ; 1 ≤ j ≤ n } and Σ Z is the a.c. spectrum of H ; – the ( v j ) ≤ j ≤ n are isolated simple eigenvalues associated to exponentially decayingeigenfunctions. FR´ED´ERIC KLOPP
It may happen that some of the gaps are closed, i.e., that the number of connected components ofΣ Z be strictly less than p . There still is a natural way to write Σ Z := σ ( H ) = p [ j =1 [ E − j , E + j ] (seesection 4.1.1), but in this case, for some j ’s, one has E + j − = E − j ; the energies E + j − = E − j , we shallcall closed gaps (see Definition 4.1). The existence of closed gaps is non generic (see [42]).The operators H • (for • ∈ { N , Z } ) admit an integrated density of states defined by(1.2) N ( E ) = lim L → + ∞ { eigenvalues of ( − ∆ + V ) | J − L,L K ∩• in ( −∞ , E ] } J − L, L K ∩ • ) . Here, the restriction of − ∆ + V to J − L, L K ∩ • is taken with Dirichlet boundary conditions; this isto fix ideas as it is known that, in the limit L → + ∞ , other self-adjoint boundary conditions wouldyield the same result for the limit (1.2).The integrated density of states is the same for H N and H Z (see, e.g., [33]). It defines the distri-bution function of some probability measure on Σ Z that is real analytic on ◦ Σ Z . Let n denote thedensity of states of H N and H Z , that is, n ( E ) = dNdE ( E ). Remark 1.3.
When L gets large, as H N L tends to H N in strong resolvent sense, interesting phe-nomena for the resonances of H N L should take place near energies in Σ N .Define τ k to be the shift by k steps to the left, that is, τ k V ( · ) = V ( · + k ). Then, for ( ℓ L ) L s.t. l L → + ∞ and L − ℓ L → + ∞ when L → + ∞ , τ ∗ l L H Z L τ l L tend to H Z in strong resolvent sense. Thus,interesting phenomena for the resonances of H Z L should take place near energies in Σ Z .1.2.1. Resonance free regions.
We start with a description of resonance free regions near the realaxis. Therefore, we introduce some operators on the positive and the negative half-lattice.Above we have defined H N ; we shall need another auxiliary operator. On ℓ ( Z − ) (where Z − = { n ≤ } ), consider the operator H − k = − ∆ + τ k V with Dirichlet boundary condition at 0 (where τ k is defined to be the shift by k steps to the left, that is, τ k V ( · ) = V ( · + k )). Let Σ − k = σ ( H − k ).As is the case for H N , one knows that σ ess ( H − k ) = Σ Z and that σ ess ( H − k ) is purely absolutelycontinuous (see, e.g., [39, Chapter 7]). H − k may also have discrete eigenvalues in R \ Σ Z .We prove Theorem 1.2.
Let I be a compact interval in ( − , . Then, (1) if I ⊂ R \ Σ N (resp. I ⊂ R \ Σ Z ), then, there exists c > such that, for L sufficiently large, H N L (resp. H Z L ) has no resonances in the rectangle { Re z ∈ I, Im z ∈ [ − c, } ; (2) if I ⊂ Σ Z , then, there exists c > such that, for L sufficiently large, H N L and H Z L have noresonances in the rectangle { Re z ∈ I, Im z ∈ [ − c/L, } ; (3) fix ≤ k ≤ p − and assume the compact interval I to be such that { v j } = ◦ I ∩ Σ N = I ∩ Σ N and I ∩ Σ Z = ∅ ( ( v j ) j are defined in the beginning of section 1.2): (a) if I ∩ Σ − k = ∅ then, there exists c > such that, for L sufficiently large such that L ≡ k mod p , H N L has a unique resonance in the rectangle { Re z ∈ I, − c ≤ Im z ≤ } ;moreover, this resonance, say z j , is simple and satisfies Im z j ≍ − e − ρ j L and | z j − λ j | ≍ e − ρ j L for some ρ j > independent of L ; (b) if I ∩ Σ − k = ∅ then, there exists c > such that, for L sufficiently large such that L ≡ k mod p , H N L has no resonance in the rectangle { Re z ∈ I, − c ≤ Im z ≤ } . So, below the spectral interval ( − , L − . For H N L , if L ≡ k mod p , each discrete eigenvalue of H N that is not an eigenvalue of ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 7 H − k generates a resonance for H N L exponentially close to the real axis (when L is large). When theeigenvalue of H − k is also an eigenvalue of H N = H +0 , it may also generate a resonance but onlymuch further away in the complex plane, at least at a distance of order 1 to the real axis.In case (3)(a) of Theorem 1.2, one can give an asymptotic expansion for the resonances (see sec-tion 5.2.1).We now turn to the description of the resonances of H • L near [ − , Some auxiliary functions. To H − k defined above, we associate N − k , the distribution func-tion of its spectral measure (that is a probability measure), i.e., for ϕ ∈ C ∞ ( R ), we define Z R ϕ ( λ ) dN − k ( λ ) := ϕ ( H − k )(0 ,
0) where ( ϕ ( H − k )( x, y )) ( x,y ) ∈ ( Z − ) denotes the kernel of the opera-tor ϕ ( H − k ).On ◦ Σ Z , the spectral measure dN − k admits a density with respect to the Lebesgue measure, say, n − k ,and this density is real analytic (see Proposition 5.1).For E ∈ ◦ Σ Z , define(1.3) S − k ( E ) := p.v. (cid:18)Z R dN − k ( λ ) λ − E (cid:19) = lim ε ↓ (cid:18)Z E − ε −∞ dN − k ( λ ) λ − E − Z + ∞ E + ε dN − k ( λ ) λ − E (cid:19) . The existence and analyticity of the Cauchy principal value S − k on ◦ Σ Z is guaranteed by the ana-lyticity of n − k (see, e.g., [18]). Moreover, for E ∈ ◦ Σ Z , one has(1.4) S − k ( E ) = lim ε → + Z R dN − k ( λ ) λ − E − iε − iπn − k ( E ) . In the lower half-plane { Im E < } , define the function(1.5) Ξ − k ( E ) := Z R dN − k ( λ ) λ − E + e − i arccos( E/ = Z R dN − k ( λ ) λ − E + E/ p ( E/ − • in the first formula, the function z arccos z is the analytic continuation to the lowerhalf-plane of the determination taking values in [ − π,
0] on the interval [ − , • in the second formula, the branch of the square root z
7→ √ z − z ∈ ( − , − k is analytic in { Im E < } and in a neighborhood of ( − , ∩ ◦ Σ Z . Moreover, Ξ − k vanishes identically if and only if V ≡ V
0. In this case, in { Im E < } and on ( − , ∩ ◦ Σ Z , the analyticfunction Ξ − k has only finitely many zeros, each of finite multiplicity (see Proposition 5.2).We shall need the analogues of the above defined functions the already introduced operator H +0 := H N = − ∆ + V considered on ℓ ( N ) with Dirichlet boundary conditions at 0. We define thefunction N +0 as the distribution function of the spectral measure of H +0 , i.e., for ϕ ∈ C ∞ ( R ), wedefine Z R ϕ ( λ ) dN +0 ( λ ) := ϕ ( H +0 )(0 , n − k , S − k and Ξ − k from H − k , one can define n +0 , S +0 and Ξ +0 from H +0 . They also satisfy Proposition 5.1, relation (1.4) andProposition 5.2. FR´ED´ERIC KLOPP
For the description of the resonances, it will be convenient to define the following functions on ◦ Σ Z (1.6) c N ( E ) := i + Ξ − k ( E ) π n − k ( E ) = 1 π n − k ( E ) (cid:16) S − k ( E ) + e − i arccos( E/ (cid:17) and(1.7) c Z ( E ) := (cid:0) S +0 ( E ) + e − i arccos( E/ (cid:1) (cid:0) S − k ( E ) + e − i arccos( E/ (cid:1) n +0 ( E ) n − k ( E ) − π π (cid:0) S +0 ( E ) + e − i arccos( E/ (cid:1) n +0 ( E ) + π (cid:0) S − k ( E ) + e − i arccos( E/ (cid:1) n − k ( E ) . We shall see that the the zeros of c • − i play a special role for the resonances of H • L : therefore, wedefine(1.8) D • = (cid:26) z ∈ ◦ Σ Z ; c • ( z ) = i (cid:27) The set D introduced in Theorem 0.1 is the set D Z ∩ ( − , Remark 1.4.
Before describing the resonances, let us explain why the operators H +0 and H − k naturally occur in this study. They respectrively are the strong resolvent limits (when L → + ∞ s.t. L ∈ p N + k ) of the operator H Z L restricted to J , L K with Dirichlet boundary conditions at 0and L “seen” respectively from the left and the right hand side.Indeed, define H L to be the operator H N L restricted to J , L K with Dirichlet boundary conditionsat L (see Remark 1.1). Note that H L is also the operator H Z L restricted to J , L K with Dirichletboundary conditions at 0 and L .Clearly, the operator H +0 is the strong resolvent limit of H L when L → + ∞ .If ˜ τ L denotes the translation by − L that unitarily maps ℓ ( J , L K ) into ℓ ( J − L, K ), then, ˜ H L =˜ τ L H L ˜ τ ∗ L converges in the strong resolvent sense to H − k when L → + ∞ and L ≡ k mod ( p ). Indeed, τ L V = τ k V as V is p periodic.1.2.3. Description of the resonances closest to the real axis.
Let ( λ l ) ≤ l ≤ L = ( λ Ll ) ≤ l ≤ L be theeigenvalues of H L (that is, the eigenvalues of H N L or H Z L restricted to J , L K with Dirichlet boundaryconditions, see remark 1.1) listed in increasing order. They are described in Theorem 4.2; thoseaway from the edges of Σ Z are shown to be nicely interspaced points at a distance roughly L − from one another.We first state our most general result describing the resonances in a uniform way. We, then, derivetwo corollaries describing the behavior of the resonance, first, far from the set of exceptional energies D • , second, close to an exceptional energy.Pick a compact interval I ⊂ ( − , ∩ ◦ Σ Z . For • ∈ { N , Z } and λ l ∈ I , for L large, define the complexnumber(1.9) ˜ z • l = λ l + 1 π n ( λ l ) L cot − ◦ c • (cid:20) λ l + 1 π n ( λ l ) L cot − ◦ c • (cid:18) λ l − i log LL (cid:19)(cid:21) where the determination of cot − is the inverse of the determination z cot( z ) mapping [0 , π ) × (0 , −∞ ) onto C + \ { i } .Note that, by Proposition 5.3, for L sufficiently large, we know that, for any l such that λ l ∈ I ,one has Im c • (cid:18) λ l − i log LL (cid:19) ∈ (0 , + ∞ ) \ { } ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 9 and Im c • (cid:20) λ l + 1 π n ( λ l ) L cot − ◦ c • (cid:18) λ l − i log LL (cid:19)(cid:21) ∈ (0 , + ∞ ) \ { } . Thus, the formula (1.9) defines ˜ z • l properly and in a unique way. Moreover, as the zeros of E c • ( E ) − i are of finite order, one checks that(1.10) − log L . L · Im ˜ z • l . − . L · Re (cid:0) ˜ z • l +1 − ˜ z • l (cid:1) where the constants are uniform for l such that λ l ∈ I .We prove the Theorem 1.3.
Pick • ∈ { N , Z } and k ∈ { , · · · , p − } . Let E ∈ ( − , ∩ ◦ Σ Z .Then, there exists η > and L > such that, for L > L satisfying L = k mod ( p ) , for each λ l ∈ I := [ E − η , E + η ] , there exists a unique resonance of H • L , say z • l , in the rectangle (cid:20) Re (˜ z • l + ˜ z • l − )2 , Re (˜ z • l + ˜ z • l +1 )2 (cid:21) + i [ − η ,
0] ; this resonance is simple and it satisfies | z • l − ˜ z • l | . L log L . This result calls for a few comments. First, the picture one gets for the resonances can be describedas follows (see also Figure 3). As long as λ l stays away from any zero of E c • ( E ) − i , the resonancesare nicely spaced points as the following proposition proves. Proposition 1.1.
Pick • ∈ { N , Z } and k ∈ { , · · · , p − } . Let I ⊂ ( − , ∩ ◦ Σ Z be a compactinterval such that I ∩ D • = ∅ .Then, for L sufficiently large, for each λ l ∈ I , the resonance z • l admits a complete asymptoticexpansion in powers of L − and one has (1.11) z • l = λ l + 1 π n ( λ l ) L cot − ◦ c • ( λ l ) + O (cid:18) L (cid:19) where the remainder term is uniform in l . resonances 1IE Log L
Figure 3: The resonances close to the real axis in the periodic case (after rescaling their imaginaryparts by L )The proof of Proposition 1.1 actually yields a complete asymptotic expansion in powers of L − forthe resonances in this zone (see section 5.2.5).Proposition 1.1 implies Theorem 0.1: we chose • = Z , k = 0 and the set D of exceptional points inTheorem 0.1 is exactly D Z ∩ ( − , Near the zeros of E c • ( E ) − i , the resonances take a “plunge” into the lower half of the complexplane (see Figure 3) and their imaginary part becomes of order L − log L . Indeed, Theorem 1.3and (1.9) imply Proposition 1.2.
Pick • ∈ { N , Z } and k ∈ { , · · · , p − } . Let E ∈ D • be a zero of E c • ( E ) − i of order q in ( − , ∩ ◦ Σ Z .Then, for α > , for L sufficiently large, if l is such that | λ l − E | ≤ L − α , the resonance z • l satisfies (1.12) Im z • l = q π n ( λ l ) · log (cid:18) | λ l − E | + (cid:16) q log L π n ( λ l ) L (cid:17) (cid:19) L · (1 + o (1)) where the remainder term is uniform in l such that | λ l − E | ≤ L − α . When • = Z , the asymptotic (1.12) shows that there can be a “resonance” phenomenon for res-onances: when the two functions Ξ − k and Ξ +0 share a zero at the same real energy, the maximalwidth of the resonances increases; indeed, the factor in front of L − log L is proportional to themultiplicity of the zero of Ξ − k Ξ +0 .1.2.4. Description of the low lying resonances.
The resonances found in Theorem 1.3 are not nec-essarily the only ones: deeper into the lower complex plane, one may find more resonances. Theyare related to the zeros of Ξ − k when • = N and Ξ − k Ξ +0 when • = Z (see Proposition 5.3).We now study what happens below the line { Im z = − η } (see Theorem 1.3) for the resonances of H N L and H Z L .The functions Ξ − k and Ξ +0 are analytic in the lower half plane and, by Proposition 5.2, they don’tvanish in an neighborhood of − i ∞ . Hence, the functions Ξ − k and Ξ +0 have only finitely many zerosin the lower half plane.We prove Theorem 1.4.
Pick • ∈ { N , Z } and k ∈ { , · · · , p − } . Let ( E • j ) ≤ j ≤ J be the zeros of E c • ( E ) − i in I + i ( −∞ , . Pick E ∈ ( − , ∩ ◦ Σ Z .There exists η > such that, for I = E + [ − η , η ] , for L sufficiently large s.t. L ≡ k mod ( p ) ,one has, • if E
6∈ { Re E • j ; 1 ≤ j ≤ J } , then, in the rectangle I + i ( −∞ , , the only resonances of H N L and H Z L are those given by Theorem 1.3; • if E ∈ { Re E • j ; 1 ≤ j ≤ J } , then, – in the rectangle I + i [ − η , , the only resonances of H N L and H Z L are those given byTheorem 1.3; – in the strip I + i [ −∞ , − η ] , the resonances of H • L are contained in J [ j =1 D (cid:0) E • j , e − η L (cid:1) – in D (cid:0) E • j , e − η L (cid:1) , the number of resonances (counted with multiplicity) is equal to theorder of E • j as a zero of E c • ( E ) − i . We see that the total number of resonances below a compact subset of ( − , ∩ ◦ Σ Z that do nottend to the real axis when L → + ∞ is finite. These resonances are related to the resonances of H ∞ to which we turn now.1.2.5. The half-line periodic perturbation.
Fix p ∈ N ∗ . On ℓ ( Z ), we now consider the operator H ∞ = − ∆ + V where V ( n ) = 0 for n ≥ V ( n + p ) = V ( n ) for n ≤ −
1. We prove
ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 11
Theorem 1.5.
The resolvent of H ∞ can be analytically continued from the upper half-plane through ( − , ∩ ◦ Σ Z to the lower half plane. The resulting operator does not have any poles in the lowerhalf-plane or on ( − , ∩ ◦ Σ Z .The resolvent of H ∞ can be analytically continued from the upper half-plane through ( − , \ Σ Z (resp. ◦ Σ Z \ [ − , ) to the lower half plane; the poles of the continuation through ( − , \ Σ Z (resp. ◦ Σ Z \ [ − , ) are exactly the zeros of the function E − e iθ ( E ) Z R dN − p − ( λ ) λ − E when continued fromthe upper half-plane through ( − , \ Σ Z (resp. ◦ Σ Z \ [ − , ) to the lower half-plane. Remark 1.5.
In Theorem 1.5 and below, every time we consider the analytic continuation of aresolvent through some open subset of the real line, we implicitly assume the open subset to benon empty.In figure 4, to illustrate Theorem 1.5, assuming that Σ Z (in blue) has a single gap that is containedin ( − , H ∞ and the presence orabsence of resonances for the different continuations . Using the same arguments as in the proof of Σ Z Figure 4: The analytic continuation of the resolvent and resonances for H ∞ Proposition 5.2, one easily sees that the continuations of the function E − e iθ ( E ) Z R dN − p − ( λ ) λ − E to the lower half plane through ( − , \ Σ Z and ◦ Σ Z \ [ − ,
2] have at most finitely many zeros andthat these zeros are away from the real axis.This also implies that the spectrum on H ∞ in [ − , ∪ Σ Z is purely absolutely continuous exceptpossibly at the points of ∂ Σ Z ∪ {− , } where ∂ Σ Z is the set of edges of Σ Z .1.3. The random case.
We now turn to the random case. Let V = V ω where ( V ω ( n )) n ∈ Z arebounded independent and identically distributed random variables. Assume that the common lawof the random variables admits a bounded compactly supported density, say, g .Set H N ω = − ∆ + V ω on ℓ ( N ) (with Dirichlet boundary condition at 0 to fix ideas). Let σ ( H N ω )be the spectrum of H N ω . Consider also H Z ω = − ∆ + V ω acting on ℓ ( Z ). Then, one knows (see,e.g., [19]) that, ω almost surely,(1.13) σ ( H Z ω ) = Σ := [ − ,
2] + supp g. One has the following description for the spectra σ ( H N ω ) and σ ( H Z ω ): • ω -almost surely, σ ( H Z ω ) = Σ; the spectrum is purely punctual; it consists of simple eigenval-ues associated to exponentially decaying eigenfunctions (Anderson localization, see, e.g., [33,19]); one can prove that, under the assumptions made above, the whole spectrum is dy-namically localized (see, e.g., [10] and references therein); • for H N ω (see, e.g., [33, 7]), one has, ω -almost surely, σ ( H N ω ) = Σ ∪ K ω where – Σ is the essential spectrum of H N ω ; it consists of simple eigenvalues associated to expo-nentially decaying eigenfunctions; – the set K ω is the discrete spectrum of H N ω ; it may be empty and depends on ω .1.3.1. The integrated density of states and the Lyapunov exponent.
It is well known (see, e.g., [33])that the integrated density of states of H , say, N ( E ) is defined as the following limit(1.14) N ( E ) = lim L → + ∞ { eigenvalues of H Z ω | J − L,L K in ( −∞ , E ] } L + 1 . The above limit does not depend on the boundary conditions used to define the restriction H Z ω | J − L,L K .It defines the distribution function of a probability measure supported on Σ. Under our assumptionson the random potential, N is known to be Lipschitz continuous ([33, 19]). Let n ( E ) = dNdE ( E ) beits derivative; it exists for almost all energies. If one assumes more regularity on g the density ofthe random variables ( ω n ) n , then, the density of states n can be shown to exist everywhere and tobe regular (see, e.g., [10]).One also defines the Lyapunov exponent, say ρ ( E ) as follows ρ ( E ) := lim L → + ∞ log k T L ( E, ω ) k L + 1where(1.15) T L ( E ; ω ) := (cid:18) E − V ω ( L ) −
11 0 (cid:19) × · · · × (cid:18) E − V ω (0) −
11 0 (cid:19)
For any E , ω -almost surely, the Lyapunov exponent is known to exist and to be independent of ω (see, e.g., [10, 33, 7]). It is positive at all energies. Moreover, by the Thouless formula [10], it ispositive and continuous for all E and it is the harmonic conjugate of n ( E ).For • ∈ { N , Z } , we now define H • ω,L to be the operator − ∆ • + V ω J ,L K . The goal of the nextsections is to describe the resonances of these operators in the limit L → + ∞ .As in the case of a periodic potential V , the resonances are defined as the poles of the analyticcontinuation of z ( H • ω,L − z ) − from C + through ( − ,
2) (see Theorem 1.1).1.3.2.
Resonance free regions.
We again start with a description of the resonance free region neara compact interval in ( − , H • ω,L -resonance free regionbelow a given energy will depend on whether this energy belongs to σ ( H • ω ) or not. We prove Theorem 1.6.
Fix • ∈ { N , Z } . Let I be a compact interval in ( − , . Then, ω -a.s., one has (1) for • ∈ { N , Z } , if I ⊂ R \ σ ( H • ω ) , then, there exists C > such that, for L sufficiently large,there are no resonances of H • ω,L in the rectangle { Re z ∈ I, ≥ Im z ≥ − /C } ; (2) if I ⊂ ◦ Σ , then, for ε ∈ (0 , , there exists L > such that, for L ≥ L , there are noresonances of H • ω,L in the rectangle { Re z ∈ I, ≥ Im z ≥ − e − η • ρL (1+ ε ) ) } where • ρ is the maximum of the Lyapunov exponent ρ ( E ) on I • η • = ( if • = N , / if • = Z . ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 13 (3) pick v j = v j ( ω ) ∈ K ω (see the description of the spectrum of H N ω just above section 1.3.1)and assume that { v j } = ◦ I ∩ σ ( H N ω ) = I ∩ σ ( H N ω ) and I ∩ Σ = ∅ , then, there exists c > suchthat, for L sufficiently large, H N ω,L has a unique resonance in { Re z ∈ I, − c ≤ Im z ≤ } ;moreover, this resonance, say z j , is simple and satisfies Im z j ≍ − e − ρ j ( ω ) L and | z j − λ j | ≍ e − ρ j ( ω ) L for some ρ j ( ω ) > independent of L . When comparing point (2) of this result with point (2) of Theorem 1.2, it is striking that the widthof the resonance free region below Σ is much smaller in the random case (it is exponentially smallin L ) than in the periodic case (it is polynomially small in L ). This a consequence of the localizednature of the spectrum, i.e., of the exponential decay of the eigenfunctions of H • ω .1.3.3. Description of the resonances closest to the real axis.
We will now see that below the reso-nance free strip exhibited in Theorem 1.6 one does find resonances, actually, many of them. Weprove
Theorem 1.7.
Fix • ∈ { N , Z } . Let I be a compact interval in ( − , ∩ ◦ Σ . Then, (1) for any κ ∈ (0 , , ω -a.s., one has n z resonance of H • ω,L s.t. Re z ∈ I, > Im z ≥ − e − L κ o L → Z I n ( E ) dE ;(2) for E ∈ I such that n ( E ) > and λ ∈ (0 , , define the rectangle R • ( E, λ, L, ε, δ ) := ( z ∈ C ; n ( E ) | Re z − E | ≤ ε/ − e η • ρ ( E ) δL ≤ e η • ρ ( E ) λ L Im z ≤ − e − η • ρ ( E ) δL ) where η • is defined in Theorem 1.6; then, ω -a.s., one has (1.16) lim δ → + lim ε → + lim L → + ∞ n z resonances of H • ω,L in R • ( E, λ, L, ε, δ ) o L ε δ = 1 . (3) for E ∈ I such that n ( E ) > , define R •± ( E, , L, ε, δ ) = ( z ∈ C ; n ( E ) | Re z − E | ≤ ε/ − e − η • ρ ( E )(1 ± δ ) L ≤ Im z < ) ; then, ω -a.s., one has (1.17) lim δ → + lim ε → + lim L → + ∞ (cid:8) resonances in R •± ( E, , L, ε, δ ) (cid:9) L ε δ = ( if ± = − , if ± = + . (4) for c > , ω -a.s., one has (1.18) lim L → + ∞ L (cid:8) z resonances of H • ω,L in I + i (cid:0) −∞ , − e − cL (cid:3)(cid:9) = Z I min (cid:18) cρ ( E ) , (cid:19) n ( E ) dE. The striking fact is that the resonances are much closer to the real axis than in the periodic case;the lifetime of these resonances is much larger. The resonant states are quite stable with lifetimesthat are exponentially large in the width of the random perturbation. Point (4) is an integratedversion of point (2). Let us also note here that when • = Z , point (4) of Theorem 1.7 is thestatement of Theorem 0.2.Note that the rectangles R • ( E, λ, L, ε, δ ) are very stretched along the real axis; their side-length in imaginary part is exponentially small in L whereas their side-length in real part is of order 1.To understand point (2) of Theorem 1.7, rescale the resonances of H • ω,L , say, ( z • l,L ( ω )) l as follows(1.19) x • l = x • l,L ( E, ω ) = n ( E ) L · (Re z • l,L ( ω ) − E ) and y • l = y • l,L ( E, ω ) = − η • ρ ( E ) L log | Im z • l,L ( ω ) | . For λ ∈ (0 , R • ( E, λ, L, ε, δ ) into {| x | ≤ Lε/ , | y − λ | ≤ δ/ } ;and the rectangles R •± ( E, , L, ε, δ ) are respectively mapped into {| x | ≤ Lε/ , ∓ δ ≤ y } . Thedenominator of the quotient in (1.16) is just the area of the rescaled R • ( E, λ, L, ε, δ ) for λ ∈ (0 , R • + ( E, , L, ε, δ ) \ R •− ( E, , L, ε, ε and δ small and L large, the rescaled resonances become uniformly distributed in the rescaled rectangles.We see that the structure of the set of resonances is very different from the one observed in theperiodic case (see Fig. 2). We will now zoom in on the resonance even more so as to make thisstructure clearer. Therefore, we consider the two-dimensional point process ξ • L ( E, ω ) defined by(1.20) ξ • L ( E, ω ) = X z • l,L resonance of H • ω,L δ ( x • l ,y • l ) where x • l , and y • l are defined by (1.19).We prove Theorem 1.8.
Fix E ∈ ( − , ∩ ◦ Σ such that n ( E ) > . Then, the point process ξ • L ( E, ω ) convergesweakly to a Poisson process in R × (0 , with intensity . That is, for any p ≥ , if ( I n ) ≤ n ≤ p resp. ( C n ) ≤ n ≤ p , are disjoint intervals of the real line R resp. of [0 , , then lim L → + ∞ P ω ; ( j ; x • l,L ( E, ω ) ∈ I y • l,L ( E, ω ) ∈ C ) = k ... ... ( j ; x • l,L ( E, ω ) ∈ I p y • l,L ( E, ω ) ∈ C p ) = k p = p Y n =1 e − µ n ( µ n ) k n k n ! , where µ n := | I n || C n | for ≤ n ≤ p . This is the analogue of the celebrated result on the Poisson structure of the eigenvalues and local-ization centers of a random system (see, e.g., [32, 31, 13]).When considering the model for • = Z , Theorem 1.8 is Theorem 0.3.In [22], we proved decorrelation estimates that can be used in the present setting to prove Theorem 1.9.
Fix E ∈ ( − , ∩ ◦ Σ and E ′ ∈ ( − , ∩ ◦ Σ such that E = E ′ , n ( E ) > and n ( E ′ ) > . Then, the limits of the processes ξ • L ( E, ω ) and ξ • L ( E ′ , ω ) are stochastically independent. Due to the rescaling, the above results give only a picture of the resonances in a zone of the type(1.21) E + L − (cid:2) − ε − , ε − (cid:3) − i h e − η • (1+ ε ) ρ ( E ) L , e − εη • ρ ( E ) L i for ε > L gets large, this rectangle is of a very small width and located very close to the real axis.Theorems 1.7, 1.8 and 1.9 describe the resonances lying closest to the real axis. As a comparisonbetween points (1) and (2) in Theorem 1.7 shows, these resonances are the most numerous. ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 15
One can get a number of other statistics (e.g. the distribution of the spacings between the reso-nances) using the techniques developed for the study of the spectral statistics of a random systemin the localized phase (see [14, 13, 21]) combined with the analysis developed in section 6.1.3.4.
The description of the low lying resonances.
It is natural to question what happens deeperin the complex plane. To answer this question, fix an increasing sequence of scales ( ℓ L ) L such that(1.22) ℓ L log L → L → + ∞ + ∞ and ℓ L L → L → + ∞ . We first show that there are only few resonances below the line { Im z = e − ℓ L } , namely Theorem 1.10.
Pick ( ℓ L ) L a sequence of scales satisfying (1.22) and I as above. ω almost surely, for L large, one has (1.23) n z resonances of H • ω,L in n Re z ∈ I, Im z ≤ − e − ℓ L oo = O ( ℓ L ) . As we shall show now, after proper rescaling, the structure of theses resonances is the same as thatof the resonances closer to the real axis.Fix E ∈ I so that n ( E ) >
0. Recall that ( z • l,L ( ω )) l be the resonances of H ω,L . We now rescalethe resonances using the sequence ( ℓ L ) L ; this rescaling will select resonances that are further awayfrom the real axis. Define(1.24) x • l = x • l,ℓ L ( ω ) = n ( E ) ℓ L (Re z • l,L ( ω ) − E ) and y • j = y • l,ℓ L ( ω ) = 12 η • ℓ L ρ ( E ) log | Im z • l,L ( ω ) | . Consider now the two-dimensional point process(1.25) ξ • L,ℓ ( E, ω ) = X z • l,L resonance of H • ω,L δ ( x • l,ℓL ,y • l,ℓL ) . We prove the following analogue of the results of Theorems 1.7, 1.8 and 1.9 for resonances lyingfurther away from the real axis.
Theorem 1.11.
Fix E ∈ ( − , ∩ ◦ Σ and E ′ ∈ ( − , ∩ ◦ Σ such that E = E ′ , n ( E ) > and n ( E ′ ) > . Fix a sequence of scales ( ℓ L ) L satisfying (1.22) . Then, one has (1) for λ ∈ (0 , , ω -almost surely lim δ → + lim ε → + lim L → + ∞ n z resonances of H • ω,L in R • ( E, λ, ℓ L , ε, δ ) o ℓ L ε δ = 1 where R • ( E, λ, L, ε, δ ) is defined in Theorem 1.7; (2) the point processes ξ • L,ℓ ( E, ω ) and ξ • L,ℓ ( E ′ , ω ) converge weakly to Poisson processes in R × (0 , + ∞ ) of intensity ; (3) the limits of the processes ξ • L,ℓ ( E, ω ) and ξ • L,ℓ ( E ′ , ω ) are stochastically independent. Point (1) shows that, in (1.23), one actually has n z resonances of H • ω,L in n Re z ∈ I, Im z ≤ − e − ℓ L oo ≍ ℓ L . Notice also that the effect of the scaling (1.24) is to select resonances that live in the rectangle E + ℓ − L (cid:2) − ε − , ε − (cid:3) − i h e − η • (1+ ε ) ρ ( E ) ℓ L , e − εη • ρ ( E ) ℓ L i This rectangle is now much further away from the real axis than the one considered in section 1.3.3.Modulo rescaling, the picture one gets for resonances in such rectangles is the same one got above in the rectangles (1.21). This description is valid almost all the way from distances to the real axisthat are exponentially small in L up to distances that are of order e − (log L ) α , α > Deep resonances.
One can also study the resonances that are even further away from the realaxis in a way similar to what was done in the periodic case in section 1.2.4. Define the followingrandom potentials on N and Z (1.26) ˜ V N ω,L ( n ) = ( ω L − n for 0 ≤ n ≤ L L + 1 ≤ n and˜ V Z ω, ˜ ω,L ( n ) = n ≤ − ω n for 0 ≤ n ≤ [ L/ ω L − n for [ L/
2] + 1 ≤ n ≤ L L + 1 ≤ n where ω = ( ω n ) n ∈ N and ˜ ω = (˜ ω n ) n ∈ N are i.i.d. and satisfy the assumptions of the beginning ofsection 1.3.Consider the operators • ˜ H N ω,L = − ∆ + ˜ V N ω,L on ℓ ( N ) with Dirichlet boundary condition at 0, • ˜ H Z ω, ˜ ω,L = − ∆ + ˜ V Z ω, ˜ ω,L on ℓ ( Z ).Clearly, the random operator ˜ H N ω,L (resp. ˜ H Z ω,L ) has the same distribution as H N ω,L (resp. H Z ω,L ).Thus, for the low lying resonances, we are now going to describe those of ˜ H N ω,L (resp. ˜ H Z ω,L ) insteadof those of H N ω,L (resp. H Z ω,L ). Remark 1.6.
The reason for this change of operators is the same as the one why, in the case of theperiodic potential, we had to distinguish various auxiliary operators depending on the congruenceof L modulo p , the period : this gives a meaning to the limiting operators when L → + ∞ .Define the probability measure dN ω ( λ ) using its Borel transform by, for Im z = 0,(1.27) Z R dN ω ( λ ) λ − z := h δ , ( H N ω − E ) − δ i . Consider the function(1.28) Ξ ω ( E ) = Z R dN ω ( λ ) λ − E + e − i arccos( E/ = Z R dN ω ( λ ) λ − E + E/ p ( E/ − z arccos z and z
7→ √ z − ω is the analogue of Ξ − k in the periodic case. One proves the analogue ofProposition 5.2 Proposition 1.3. If ω = 0 , one has Ξ ω ( E ) ∼ | E |→∞ Im E< − ω E − . Thus, ω almost surely, Ξ ω does notvanish identically in { Im E < } .Pick I ⊂ ◦ Σ ∩ ( − , compact. Then, ω almost surely, the number of zeros of Ξ ω (counted withmultiplicity) in I + i ( −∞ , ε ] is asymptotic to Z I n ( E ) ρ ( E ) dE | log ε | as ε → + ; moreover, ω almostsurely, there exists ε ω > such that all the zeroes of Ξ ω in I + i [ − ε ω , are simple. It seems reasonable to believe that, except for the zero at − i ∞ , ω almost surely, all the zeros ofΞ ω are simple; we do not prove itFor the “deep” resonances, we then prove ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 17
Theorem 1.12.
Fix I ⊂ ◦ Σ ∩ ( − , a compact interval. There exists c > such that, withprobability 1, there exists c ω > such that, for L sufficiently large, one has (1) for each resonance of ˜ H N ω,L (resp. ˜ H Z ω, ˜ ω,L ) in I + i (cid:0) −∞ , − e − cL (cid:3) , say E , there exists a uniquezero of Ξ ω (resp. Ξ ω Ξ ˜ ω ), say ˜ E , such that | E − ˜ E | ≤ e − c ω L ; (2) reciprocally, to each zero (counted with multiplicity) of Ξ ω (resp. Ξ ω Ξ ˜ ω ) in the rectangle I + i (cid:0) −∞ , − e − cL (cid:3) , say ˜ E , one can associate a unique resonance of ˜ H N ω,L (resp. ˜ H Z ω, ˜ ω,L ),say E , such that | E − ˜ E | ≤ e − c ω L . One can combine this result with the description of the asymptotic distribution of the resonancesgiven by Theorem 1.11 to obtain the asymptotic distributions of the zeros of the function Ξ ω neara point E − iε when ε → + . Indeed, let ( z l ( ω )) l be the zeros of Ξ ω in { Im E < } . Rescale thezeros:(1.29) x l,ε ( ω ) = n ( E ) | log ε | · (Re z l ( ω ) − E ) and y l,ε ( ω ) = − ρ ( E ) | log ε | log | Im z l ( ω ) | and consider the two-dimensional point process ξ ε ( E, ω ) defined by(1.30) ξ ε ( E, ω ) = X z l ( ω ) zeros of Ξ ω δ ( x l,ε ,y l,ε ) . Then, one has
Corollary 1.1.
Fix E ∈ I such that n ( E ) > . Then, the point process ξ ε ( E, ω ) converges weaklyto a Poisson process in R × R with intensity . The function Ξ ω has been studied in [26, 27] where the average density of its zeros was computed.Here, we obtain a more precise result.1.3.6. The half-line random perturbation. On ℓ ( Z ), we now consider the operator H ∞ ω = − ∆ + V ω where V ω ( n ) = 0 for n ≥ V ω ( n ) = ω n for n ≤ − ω n ) n ≥ are i.i.d. and have the samedistribution as above. Recall that Σ is the almost sure spectrum of H Z ω (on ℓ ( Z )). We prove Theorem 1.13.
First, ω almost surely, the resolvent of H ∞ ω does not admit an analytic continua-tion from the upper half-plane through ( − , ∩ ◦ Σ to any subset of the lower half plane. Nevertheless, ω -almost surely, the spectrum of H ∞ ω in ( − , ∩ ◦ Σ is purely absolutely continuous.Second, ω almost surely, the resolvent of H ∞ ω does admit a meromorphic continuation from theupper half-plane through ( − , \ Σ to the lower half plane; the poles of this continuation are ex-actly the zeros of the function E − e iθ ( E ) Z R dN ω ( λ ) λ − E when continued from the upper half-planethrough ( − , \ Σ to the lower half-plane.Third, ω almost surely, the spectrum of H ∞ ω in ◦ Σ \ [ − , is pure point associated to exponentiallydecaying eigenfunctions; hence, the resolvent of H ∞ ω cannot be be continued through ◦ Σ \ [ − , . In figure 5, to illustrate Theorem 1.13, assuming that Σ Z (in blue) has a single gap that is containedin ( − , H ∞ ω and the associated resonances;we also indicate the real intervals of spectrum through which the the resolvent of H ∞ ω does notadmit an analytic continuation and the spectral type of H ∞ ω in the intervals. Let us also note herethat if 0 ∈ supp g (where g is the density of the random variables defining the random potential),then, by (1.13), one has [ − , ⊂ Σ. In this case, there is no possibility to continue the resolventof H ∞ ω to the lower half plane passing through [ − , H ∞ , when continued through Σ Z -2 2no analytic cont.but a.c. spectrumres.and dense p.p. spectrumno analytic cont. Figure 5: The analytic continuation of the resolvent and resonances for H ∞ ω ( − , ∩ ◦ Σ, the operator H ∞ ω does not have any resonances but for very different reasons.When one does the continuation through ( − , \ Σ, one sees that the number of resonances isfinite; “near” the real axis, the continuation of the function E − e iθ ( E ) Z R dN ω ( λ ) λ − E has nontrivial imaginary part and near ∞ it does not vanish.Theorem 1.13 also shows that the equation studied in [26, 27], i.e., the equation Ξ ω ( E ) = 0, doesnot describe the resonances of H ∞ ω as is claimed in these papers: these resonances do not existas there is no analytic continuation of the resolvent of H ∞ ω through ( − , ∩ Σ! As is shown inTheorem 1.12, the solutions to the equation Ξ ω ( E ) = 0 give an approximation to the resonancesof H N ω,L (see Theorem 1.12).1.4. Outline of and reading guide to the paper.
In the present section, we shall explain themain ideas leading to the proofs of the results presented above.In section 2, we prove Theorem 1.1; this proof is classical. As a consequence of the proof, one seesthat, in the case of the half-lattice N (resp. lattice Z ), the resonances are the eigenvalues of a rankone (resp. two) perturbation of ( − ∆ + V ) | J ,L K with Dirichlet b.c. The perturbation depends inan explicit way on the resonance. This yields a closed equation for the resonances in terms of theeigenvalues and normalized eigenfunctions of the Dirichlet restriction ( − ∆ + V ) | J ,L K . To obtain adescription of the resonances we then are in need of a “precise” description of the eigenvalues andnormalized eigenfunctions. Actually the only information needed on the normalized eigenfunctionsis their weight at the point L (and the point 0 in the full lattice case), 0 and L being the endpointsof J , L K .In section 3, we solve the two equations obtained previously under the condition that the weightof the normalized eigenfunctions at L (and 0) be much smaller than the spacing between theDirichlet eigenvalues. This condition entails that the resonance equation we want to solve essentiallyfactorizes and become very easy to solve (see Theorems 3.1, 3.2 and 3.3), i.e., it suffices to solve itnear any given Dirichlet eigenvalue.For periodic potentials, the condition that the eigenvalue spacing is much larger than the weightof the normalized eigenfunctions at L (and 0) is not satisfied: both quantities are of the same orderof magnitude (see Theorem 4.2) for the Dirichlet eigenvalues in the bulk of the spectrum, i.e., thevast majority of them. This is a consequence of the extended nature of the eigenfunctions in thiscase. Therefore, we find another way to solve the resonance equation. This way goes through amore precise description of the Dirichlet eigenvalues and normalized eigenfunctions which is thepurpose of Theorems 4.2. We use this description to reduce the resonance equation to an effective ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 19 equation (see Theorem 5.1) up to errors of order O ( L −∞ ). It is important to obtain errors of atmost that size. Indeed, the effective equation may have solutions to any order (the order is finiteand only depends on V but it is unknown); thus, to obtain solutions to the true equation fromsolutions to the effective equation with a good precision, one needs the two equations to differ byat most O ( L −∞ ). We then solve the effective equation and, in section 5.2, prove the results ofsection 1.2.On the other hand, for random potentials, it is well known that the eigenfunctions of the Dirichletrestriction ( − ∆ + V ) | J ,L K are exponentially localized and, for most of them localized, far from theedge of J , L K . Thus, their weight at L (and 0 in the full lattice case) is typically exponentiallysmall in L ; the eigenvalue spacing however is typically of order L − . We can then use the resultsof section 3 to solve the resonance equation. The real part of a given resonance is directly relatedto a Dirichlet eigenvalue and its imaginary part to the weight of the corresponding eigenfunctionat L (and 0 in the full lattice case). The main difficulty is to find the asymptotic behavior of thisweight. Indeed, while it is known that, in the random case, eigenfunctions decay exponentiallyaway from a localization center and while it is known that, for the full random Hamiltonian (i.e.the Hamiltonian on the line or half-line with a random potential), at infinity, this decay rate isgiven by the Lyapunov exponent, to the best of our knowledge, before the present work, it was notknown at which length scale this Lyapunov behavior sets in (with a good probability). Answeringthis question is the purpose of Theorems 6.2 and 6.3 proved in section 6.3: we show that, for the1-dimensional Anderson model, for δ > L sufficiently large, all theeigenfunctions exhibit an exponential decay (we obtain both an upper and a lower bound on theeigenfunctions) at a rate equal to the Lyapunov exponent at the corresponding energy (up to anerror of size δ ) as soon as one is at a distance δL from the corresponding localization center.These bounds give estimates on the weight of most eigenfunctions at the point L (and 0 in thefull lattice case): it is directly related to the distance of the corresponding localization center tothe points L (and 0). One can then transform the known results on the statistics of the (rescaled)eigenvalues and (rescaled) localization centers into statistics of the (rescaled) resonances. This isdone in section 6.2 and proves most of the results in section 1.3.Finally, section 6.4 is devoted to the study of the full line Hamiltonian obtained from the freeHamiltonian on one half-line and a random Hamiltonian on the other half-line; it contains inparticular the proof of Theorem 1.13.2. The analytic continuation of the resolvent
Resonances for Jacobi matrices were considered in various works (see, e.g., [6, 16] and referencestehrein). For the sake of completeness, we provide an independent proof of Theorem 1.1. It followsstandard ideas that were first applied in the continuum setting, i.e., for partial differential operatorsinstead of finite difference operators (see, e.g., [38] and references therein).The proof relies on the fact that the resolvent of free Laplace operator can be continued holomor-phically from C + to C \ (( −∞ , − ∪ [2 , + ∞ )) as an operator valued function from l to l .This is an immediate consequence of the fact that, by discrete Fourier transformation, − ∆ is theFourier multiplier by the function θ θ .Indeed, for − ∆ on ℓ ( Z ) and Im E >
0, one has, for ( n, m ) ∈ Z (assume n − m ≥ h δ n , ( − ∆ − E ) − δ m i = 12 π Z π e − i ( n − m ) θ θ − E dθ = 12 iπ Z | z | =1 z n − m z − Ez + 1 dz = 12 p ( E/ − (cid:16) E/ − p ( E/ − (cid:17) n − m = e i ( n − m ) θ ( E ) sin θ ( E )(2.1) where E = 2 cos θ ( E ) and the determination θ = θ ( E ) is chosen so that Im θ > θ ∈ ( − π, E >
0. The determination satisfies θ (cid:0) E (cid:1) = θ ( E ).The map E θ ( E ) can continued analytically from C + to the cut plane C \ (( −∞ , − ∪ [2 , + ∞ ))as shown in Figure 6. E −π θ −2 Figure 6: The mapping E θ ( E )The continuation is one-to-one and onto from C \ (( −∞ , − ∪ [2 , + ∞ )) to ( − π,
0) + i R . It definesa determination of E arccos( E/
2) = θ ( E ).Clearly, using (2.1), this continuation yields an analytic continuation of R Z := ( − ∆ − E ) − from { Im E > } to C \ (( −∞ , − ∪ [2 , + ∞ )) as an operator from l to l .Let us now turn to the half-line operator, i.e., − ∆ on N with Dirichlet condition at 0. Pick E suchthat Im E > E = 2 cos θ where the determination θ = θ ( E ) is chosen as above. If for v ∈ C N bounded and n ≥ −
1, one sets v − = 0 and(2.2) [ R N ( E )( v )] n = 12 i sin θ ( E ) n X j = − v j · sin(( n − j ) θ ( E )) − e iθ ( E ) sin(( n + 1) θ ( E ))2 i sin θ ( E ) X j ≥ e ijθ ( E ) v j . Then, for Im
E >
0, a direct computations shows that(1) for v ∈ ℓ ( N ), the vector R N ( E )( v ) is in the domain of the Dirichlet Laplacian on ℓ ( N ) ,i.e., [ R N ( E )( v )] − = 0;(2) for n ≥
0, one checks that(2.3) [ R N ( E )( v )] n +1 + [ R N ( E )( v )] n − − E [ R N ( E )( v )] n = v n . (3) R N ( E ) defines a bounded map from ℓ ( N ) to itself;Thus, R N ( E ) is the resolvent of the Dirichlet Laplacian on N at energy E for Im E >
0. Using thecontinuation of E θ ( E ), formula (2.2) yields an analytic continuation of the resolvent R N ( E ) asan operator from l to l . Remark 2.1.
Note that the resolvent R N ( E ) at an energy E s.t. Im E < θ ( E ) is replaced by − θ ( E ). For (2.2), one has to assume that ( v j ) j ∈ N decays fast enough at ∞ .To deal with the perturbation V , we proceed in the same way on Z and on N . Set V L = V J ,L K (seen as a function on N or Z depending on the case). Letting R ( E ) be either R Z ( E ) or R N ( E ),we compute − ∆ + V L − E = ( − ∆ − E )(1 + R ( E ) V L ) = (1 + V L R ( E ))( − ∆ L − E ) . Thus, it suffices to check that the operator R ( E ) V L (resp. V L R ( E )) can be analytically continuedas an operator from l to l (resp. l to l ). This follows directly from (2.2) and the fact V L has finite rank.To complete the proof of Theorem 1.1, we just note that, as • E R ( E ) V L (resp. E V L R ( E )) is a finite rank operator valued function analytic onthe connected set C \ (( −∞ , − ∪ [2 , + ∞ )), ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 21 • − R ( E ) V L (resp. V L R ( E )) for Im E > E for which − R ( E ) V L (resp. V L R ( E )) is discrete. Hence, the set of resonances is discrete.This completes the proof of the first part of Theorem 1.1. To prove the second part, we will firstwrite a characteristic equation for resonances. The bound on the number of resonances will thenbe obtained through a bound on the number of solutions to this equation.2.1. A characteristic equation for resonances.
In the literature, we did not find a character-istic equation for the resonances in a form suitable for our needs. The characteristic equation wederive will take different forms depending on whether we deal with the half-line or the full lineoperator. But in both cases, the coefficients of the characteristic equation will be constructed fromthe spectral data (i.e. the eigenvalues and eigenfunctions) of the operator H L (see Remark 1.4).2.2. In the half-line case.
We first consider H N L on ℓ ( N ) and prove Theorem 2.1.
Consider the operator H L defined as H N L restricted to J , L K with Dirichlet boundaryconditions at L and define • ( λ j ) ≤ j ≤ L = ( λ j ( L )) ≤ j ≤ L are the Dirichlet eigenvalues of H N L ordered so that λ j < λ j +1 ; • a N j = a N j ( L ) = | ϕ j ( L ) | where ϕ j = ( ϕ j ( n )) ≤ n ≤ L is a normalized eigenvector associated to λ j .Then, an energy E is a resonance of H N L if and only if (2.4) S L ( E ) := L X j =0 a N j λ j − E = − e − iθ ( E ) , E = 2 cos θ ( E ) , the determination of θ ( E ) being chosen so that Im θ ( E ) > and Re θ ( E ) ∈ ( − π, when Im E > . Let us note that(2.5) ∀ ≤ j ≤ L, a N j ( L ) > L X j =0 a N j ( L ) = L X j =0 | ϕ j ( L ) | = 1 . Proof of Theorem 2.1.
By the proof of the first statement of Theorem 1.1 (see the beginning ofsection 2), we know that an energy E is a resonance if and only if − R ( E ) V L where R ( E ) is defined by (2.2). Pick E an resonance and let u = ( u n ) n ≥ be a resonant statethat is an eigenvector of R ( E ) V L associated to the eigenvalue −
1. As V Ln = 0 for n ≥ L + 1,equation (2.2) yields that, for n ≥ L + 1, u n = βe inθ ( E ) for some fixed β ∈ C ∗ . As u = − R ( E ) V L u ,for n ≥ L + 1, it satisfies u n +1 + u n − = E u n . Thus, u L +1 = e iθ ( E ) u L and by (2.3), u is a solutionto the eigenvalues problem ( u n +1 + u n − + V n u n = E u n , ∀ n ∈ J , L K u − = 0 , u L +1 = e iθ ( E ) u L This can be equivalently be rewritten as(2.6) V · · · V V L − · · · V L + e iθ ( E ) u ... u L = E u ... u L The matrix in (2.6) is the Dirichlet restriction of H N L to J , L K perturbed by the rank one operator e iθ ( E ) δ L ⊗ δ L . Thus, by rank one perturbation theory (see, e.g., [36]), an energy E is a resonanceif and only if if satisfies (2.4).This completes the proof of Theorem 2.1. (cid:3) Let us now complete the proof of Theorem 1.1 for the operator on the half-line. Let us first notethat, for Im
E >
0, the imaginary part of the left hand side of (2.4) is positive by (2.7). On theother hand, the imaginary part of the right hand side of (2.4) is equal to − e Im θ ( E ) sin(Re θ ( E ))and, thus, is negative (recall that Re θ ( E ) ∈ ( − π,
0) (see fig. 1). Thus, as already underlined,equation (2.4) has no solution in the upper half-plane or on the interval ( − , L + 2 in thevariable z = e − iθ ( E ) (2.7) L Y k =0 (cid:0) z − λ k z + 1 (cid:1) − L X j =0 a N j Y ≤ k ≤ Lk = j (cid:0) z − λ k z + 1 (cid:1) = 0 . We are looking for the solutions to (2.7) in the upper half-plane. As the polynomial in the right handside of (2.7) has real coefficients, its zeros are symmetric with respect to the real axis. Moreover,one notices that, by (2.5), 0 is a solution to (2.7). Hence, the number of solutions to (2.7) in theupper half-plane is bounded by L . This completes the proof of Theorem 1.1.2.3. On the whole line.
Now, consider H Z L on ℓ ( Z ). We prove Theorem 2.2.
Using the notations of Theorem 2.1, an energy E is a resonance of H Z L if and onlyif (2.8) det L X j =0 λ j − E (cid:18) | ϕ j ( L ) | ϕ j (0) ϕ j ( L ) ϕ j (0) ϕ j ( L ) | ϕ j (0) | (cid:19) + e − iθ ( E ) = 0 where det ( · ) denotes the determinant of a square matrix, E = 2 cos θ ( E ) and the determination of θ ( E ) is chosen as in Theorem 2.1. So, an energy E is a resonance of H Z L if and only if − e − iθ ( E ) belongs to the spectrum of the 2 × L ( E ) := L X j =0 λ j − E (cid:18) | ϕ j ( L ) | ϕ j (0) ϕ j ( L ) ϕ j (0) ϕ j ( L ) | ϕ j (0) | (cid:19) . Proof of Theorem 2.2.
The proof is the same as that of Theorem 2.1 except that now E is aresonance if there exists u a non trivial solution to the eigenvalues problem ( u n +1 + u n − + V n u n = E u n , ∀ n ∈ J , L K u − = e iθ ( E ) u and u L +1 = e iθ ( E ) u L This can be equivalently be rewritten as V + e iθ ( E ) · · · V V L − · · · V L + e iθ ( E ) u ... u L = E u ... u L ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 23
Thus, using rank one perturbations twice, we find that an energy E is a resonance if and only if e iθ ( E ) L X j =0 | ϕ j (0) | λ j − E e iθ ( E ) L X j =0 | ϕ j ( L ) | λ j − E = e iθ ( E ) X ≤ j,j ′ ≤ L ϕ j ( L ) ϕ j ′ (0) ϕ j ′ ( L ) ϕ j (0)( λ j − E )( λ j ′ − E ) , that is, if and only is (2.8) holds. This completes the proof of Theorem 2.2. (cid:3) Let us now complete the proof of Theorem 1.1 for the operator on the full-line. Let us first showthat (2.8) has no solution in the upper half-plane. Therefore, if − e − iθ ( E ) belongs to the spectrumof the matrix defined by (2.8) and if u ∈ C is a normalized eigenvector associated to − e − iθ ( E ) , onehas L X j =0 λ j − E (cid:12)(cid:12)(cid:12)(cid:12)(cid:28)(cid:18) ϕ j ( L ) ϕ j (0) (cid:19) , u (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) = − e − iθ ( E ) . This is impossible in the upper half-plane and on ( − ,
2) as the two sides of the equation haveimaginary parts of opposite signs.Note that L X j =0 (cid:18) ϕ j ( L ) ϕ j (0) (cid:19) (cid:0) ϕ j ( L ) ϕ j (0) (cid:1) = (cid:18) (cid:19) . Note also that − e − iθ ( E ) is an eigenvalue of (2.8) if and only if it satisfies(2.10) 1 + e iθ ( E ) L X j =0 | ϕ j ( L ) | + | ϕ j (0) | λ j − E = − e iθ ( E ) X ≤ j,j ′ ≤ L (cid:12)(cid:12)(cid:12)(cid:12) ϕ j (0) ϕ j ′ (0) ϕ j ( L ) ϕ j ′ ( L ) (cid:12)(cid:12)(cid:12)(cid:12) ( λ j − E )( λ j ′ − E ) . As the eigenvalues of H L are simple, one computes(2.11) X ≤ j,j ′ ≤ L (cid:12)(cid:12)(cid:12)(cid:12) ϕ j (0) ϕ j ′ (0) ϕ j ( L ) ϕ j ′ ( L ) (cid:12)(cid:12)(cid:12)(cid:12) ( λ j − E )( λ j ′ − E ) = 2 X ≤ j ≤ L λ j − E X j ′ = j λ j ′ − λ j (cid:12)(cid:12)(cid:12)(cid:12) ϕ j (0) ϕ j ′ (0) ϕ j ( L ) ϕ j ′ ( L ) (cid:12)(cid:12)(cid:12)(cid:12) . Thus, equation (2.10) is equivalent to the following polynomial equation of degree 2( L + 1) in thevariable z = e − iθ ( E ) (2.12) z L Y k =0 (cid:0) z − λ k z + 1 (cid:1) − L X j =0 (2 a Z j z + b Z j ) Y ≤ k ≤ Lk = j (cid:0) z − λ k z + 1 (cid:1) = 0 . where we have defined a Z j := 12 (cid:0) | ϕ j ( L ) | + | ϕ j (0) | (cid:1) = 12 (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ϕ j ( L ) ϕ j (0) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) = 12 (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | ϕ j ( L ) | ϕ j (0) ϕ j ( L ) ϕ j (0) ϕ j ( L ) | ϕ j (0) | (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) . (2.13)and b Z j := X j ′ = j λ j ′ − λ j (cid:12)(cid:12)(cid:12)(cid:12) ϕ j (0) ϕ j ′ (0) ϕ j ( L ) ϕ j ′ ( L ) (cid:12)(cid:12)(cid:12)(cid:12) . The sequence ( a Z j ) j also satisfies (2.5). Taking | E | to + ∞ in (2.11), one notes that(2.14) L X j =0 b Z j = 0 and L X j =0 λ j b Z j = − X ≤ j,j ′ ≤ L (cid:12)(cid:12)(cid:12)(cid:12) ϕ j (0) ϕ j ′ (0) ϕ j ( L ) ϕ j ′ ( L ) (cid:12)(cid:12)(cid:12)(cid:12) = − . We are looking for the solutions to (2.12) in the upper half-plane. As the polynomial in the righthand side of (2.12) has real coefficients, its zeros are symmetric with respect to the real axis.Moreover, one notices that, by (2.14), 0 is a root of order two of the polynomial in (2.12). Hence,as the polynomial has degree 2 L + 3, the number of solutions to (2.12) in the upper half-plane isbounded by L . This completes the proof of Theorem 1.1.3. General estimates on resonances
By Theorems 2.1 and 2.2, we want to solve equations (2.4) and (2.8) in the lower half-plane.We first derive some general estimates for zones in the lower half-plane free of solutions to equa-tions (2.4) and (2.8) (i.e. resonant free zones for the operators H N L and H Z L ) and later a result onthe existence of solutions to equations (2.4) and (2.8) (i.e. resonances for the operators H N L and H Z L ).3.1. General estimates for resonant free regions.
We keep the notations of Theorems 2.1and 2.2. To simplify the notations in the theorems of this section, we will write a j for either a N j when solving (2.4) or a Z j when solving (2.8). We will specify the superscript only when there is riskof confusion.We first prove Theorem 3.1.
Fix δ > . Then, there exists C > (independent of V and L ) such that, for any L and j ∈ { , · · · , L } such that − δ ≤ λ j − + λ j < λ j +1 + λ j ≤ − δ , equations (2.4) and (2.8) have no solution in the set (3.1) U j := E ∈ C ; Re E ∈ (cid:20) λ j + λ j − , λ j + λ j +1 (cid:21) ≥ C · θ ′ δ Im E > − a j d j | sin Re θ ( E ) | where the map E θ ( E ) is defined in section 2 and we have set (3.2) d j := min ( λ j +1 − λ j , λ j − λ j − , and θ ′ δ := max | E |≤ − δ | θ ′ ( E ) | . (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) R~ λ j λ j+1 λ j−1 U U j j j
Figure 7: The resonance free zones U j and˜ U j .In Theorem 3.1 there are no conditions on the num-bers ( a j ) j or ( d j ) j except their being positive. In ourapplication to resonances, this holds. Theorem 3.1becomes optimal when a j ≪ d j . In our applica-tion to resonances, for periodic operators, one has a j ≍ L − and d j ≍ L − (see Theorem 5.2) and forrandom operators, one has a j ≍ e − cL and d j & L − (see Theorem 6.2 and (6.10)). Thus, in the randomcase, Theorem 3.1 will provide an optimal strip freeof resonances whereas in the periodic case we will usea much more precise computation (see Theorem 5.1)to obtain sharp results.When a j ≪ d j , one proves the existence of anotherresonant free region near a energy λ j , namely, Theorem 3.2.
Fix δ > . Pick j ∈ { , · · · , L } such that − δ < λ j − + λ j < λ j +1 + λ j < − δ .There exists C > (depending only on δ ) such that, for any L , if a j ≤ d j /C , equations (2.4) ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 25 and (2.8) have no solution in the set ˜ U j := E ∈ C ; Re E ∈ (cid:20) λ j + λ j − , λ j − Ca j (cid:21) ∪ (cid:20) λ j + Ca j , λ j + λ j +1 (cid:21) − Ca j ≤ Im E ≤ − a j d j /C [ E ∈ C ; Re E ∈ (cid:20) λ j + λ j − , λ j + λ j +1 (cid:21) − d j /C ≤ Im E ≤ − Ca j (3.3)Theorem 3.2 becomes optimal when a j is small and d j is of order one. This will be sufficient todeal with the isolated eigenvalues for both the periodic and the random potential. It will also besufficient to give a sharp description of the resonant free region for random potentials. For theperiodic potential, we will rely a much more precise computations (see Theorem 5.1).Note that Theorem 3.2 guarantees that, if d j is not too small, outside R j , resonances are quite farbelow the real axis. Proof of Theorem 3.1.
The basic idea of the proof is that, for E close to λ j , S L ( E ) and the matrixΓ L ( E ) are either large or have a very small imaginary part while, as − < λ j − + λ j < λ j +1 + λ j < e − iθ ( E ) has a large imaginary part. Thus, (2.4) and (2.8) have no solution in this region.We start with equation (2.4). Pick E ∈ U j for some C large to be chosen later on. Assume firstthat | E − λ j | ≤ a j d j (2 + C d j ) − for C := 2 e /C . Recall that 0 < a j , d j ≤
1. Note that, for C sufficiently large, for E ∈ U j , one has (cid:12)(cid:12)(cid:12) Im e − iθ ( E ) (cid:12)(cid:12)(cid:12) = e Im θ ( E ) | sin Re θ ( E ) | = e Im[ θ ( E ) − θ (Re E )] | sin Re θ ( E ) |≥ e θ ′ δ Im E | sin Re θ ( E ) | ≥ e − /C | sin Re θ ( E ) | (3.4)and(3.5) (cid:12)(cid:12)(cid:12) e − iθ ( E ) (cid:12)(cid:12)(cid:12) ≤ ≤ e /C . One estimates(3.6) | S L ( E ) | ≥ a j | λ j − E | − X k = j a k | λ k − E | ≥ d j + C − X k = j a k min k = j | λ k − λ j | ≥ C = 2 e /C . Thus, comparing (3.6) and (3.5), we see that equation (2.4) has no solution in the set U j ∩{| E − λ j | ≤ a j d j (2 + Cd j ) − } .Assume now that | E − λ j | > a j d j (2 + C d j ) − . Then, for E ∈ U j , one has(3.7) | Im E | ≤ θ ′ δ C a j d j | sin(Re θ ( E )) | . Thus, for E ∈ U j ∩ {| E − λ j | > a j d j (2 + C d j ) − } , one computes | Im S L ( E ) | ≤ | Im E | a j | λ j − Re E | + | Im E | + 4 d j + | Im E | ! ≤ θ ′ δ C a j d j | sin(Re θ ( E )) | (2 + C d j ) a j a j d j + 4 d j ! ≤ θ ′ δ C (1 + e /C ) | sin(Re θ ( E )) | ≤ e − /C | sin(Re θ ( E )) | (3.8) provided C satisfies 8 e /C (1 + e /C ) < θ ′ δ C .Hence, the comparison of (3.4) with (3.8) shows that (2.4) has no solution in U j ∩ {| E − λ j | >a j d j (2 + C d j ) − } if we choose C large enough (independent of ( a j ) j and ( λ j ) j ). Thus, we haveproved that for some C > a j ) j and ( λ j ) j ), (2.4) has no solutionin U j .Let us now turn to the case of equation (2.8). The basic ideas are the same as for equation (2.4).Consider the matrix Γ L ( E ) defined by (2.9). The summands in (2.9) are hermitian, of rank 1 andtheir norm is given by (2.13).Assume that E ∈ U j is a solution to (2.8). Define the vectors v j := a − / j (cid:18) ϕ j ( L ) ϕ j (0) (cid:19) for j ∈ { , · · · , L } . Here a j = a Z j .Note that by definition of a j , one has k v j k = 2. Pick u in C , a normalized eigenvector of Γ L ( E )associated to the eigenvalue − e − iθ ( E ) . Thus, u satisfies(3.9) L X j =0 a j h v j , u i v j λ j − E = − e − iθ ( E ) u Note that, by assumption, one has(3.10) sup E ∈ U j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k = j a k h v k , u i v k λ k − E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . d j and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Im X k = j a k |h v k , u i| λ k − E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . | Im E | d j where the constants are independent of C , the one defining U j .Taking the (real) scalar product of equation (3.9) with u , and then the imaginary part, we obtain − a j |h v j , u i| Im E | λ j − E | + Im (cid:16) e − iθ ( E ) (cid:17) = O | Im E | d j ! Thus, for E ∈ U j , as a j ≤
1, for C in (3.1) sufficiently large (depending only on δ ), a j |h v j , u i| | Im E || λ j − E | ≥ | sin(Re θ ( E )) | . Hence, for a solution to (2.8) in U j and u as above, one has | λ j − E | ≤ |h v j , u i| s a j | Im E || sin(Re θ ( E )) | ≤ s a j | Im E || sin(Re θ ( E )) | . Hence, by the definition of U j , for C large, we get(3.11) (cid:12)(cid:12)(cid:12)(cid:12) a j λ j − E (cid:12)(cid:12)(cid:12)(cid:12) ≥ Cθ ′ δ d j ≫ d j . By (3.10), the operator Γ L ( E ) can be written as(3.12) Γ L ( E ) = a j λ j − E v j ⊗ v j + R j ( E ) + iI j ( E )where R j ( E ) and I j ( E ) are self-adjoint ( I j is non negative) and satisfy(3.13) k R j ( E ) k . d j and k I j ( E ) k . | Im E | d j . ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 27
An explicit computation shows that the eigenvalues of the two-by-two matrix a j λ j − E v j ⊗ v j + R j ( E )satisfy either λ = a j λ j − E (cid:18) O (cid:18) d j Cθ ′ δ (cid:19)(cid:19) or | Im λ | . | Im E | a j where the implicit constants are independent of the one defining U j .Thus, by (3.12), using (3.11) and the second estimate in (3.13), we see that the eigenvalues of thematrix Γ L ( E ) satisfyeither λ = a j λ j − E (cid:18) O (cid:18) d j Cθ ′ δ (cid:19)(cid:19) or | Im λ | ≤ Cθ ′ δ . Clearly, for C large, no such value can be equal to − e − iθ ( E ) being to large by (3.11) in the firstcase or having too small imaginary part in the second. The proof of Theorem 3.1 is complete. (cid:3) Proof of Theorem 3.2.
Again, we start with the solutions to (2.4). For z ∈ ˜ U j , we computeIm S L ( E ) = L X k =0 a k Im E ( λ k − Re E ) + Im E = a j Im E ( λ j − Re E ) + Im E + X ≤ k ≤ Lk = j − a k Im E ( λ k − Re E ) + Im E . (3.14)When − d j /C ≤ Im E ≤ − Ca j , the second equality above and (2.5) yield, for C sufficiently large,(3.15) 0 ≤ Im S L ( E ) . a j | Im E | + | Im E | d j + Im E ≤ C .
On the other hand, for some
K >
0, one has (cid:12)(cid:12)(cid:12) Im e − iθ ( E ) (cid:12)(cid:12)(cid:12) ≥ | Im e − iθ (Re E ) | − Kd j /C. Now, as, under the assumptions of Theorem 3.2, one has(3.16) min E ∈ h λj + λj − , λj + λj +12 i (cid:12)(cid:12)(cid:12) Im e − iθ ( E ) (cid:12)(cid:12)(cid:12) ≥
14 min (cid:18)q − ( λ j + λ j − ) , q − ( λ j + λ j +1 ) (cid:19) , we obtain that (2.4) has no solution in ˜ U j ∩ {− d j /C ≤ Im E ≤ − Ca j } .Pick now E ∈ ˜ U j such that − Ca j ≤ Im E ≤ − a j d j /C . Then, (3.5) and (2.5) yield, for C sufficientlylarge, Im S L ( E ) . a j Im EC a j + Im E + Ca j d j ≤ C + 12 C .
The imaginary part of e − iθ ( E ) is estimated as above. Thus, for C sufficiently large, (2.4) has nosolution in ˜ U j ∩ {− Ca j ≤ Im E ≤ − a j d j /C } .The case of equation (2.8) is studied in exactly the same way except that, as in the proof ofTheorem 3.1, one has to replace the study of S L ( E ) by that of h Γ L ( E ) u, u i for u a normalizedeigenvector of Γ L ( E ) associated to − e − iθ ( E ) and, thus, the coefficient a k in (3.14) gets multipliedby a factor |h v k , u i| that is bounded by 2.This completes the proof of Theorem 3.2. (cid:3) The resonances near an “isolated” eigenvalue.
We will now solve equation (2.4) near agiven λ j under the additional assumptions that a j ≪ d j . By Theorems 3.1 and 3.2, we will do soin the rectangle R j (see Fig. 7). Actually, we prove that, in R j , there is exactly one resonance andgive an asymptotic for this resonance in terms of a j , d j and λ j . This result is going to be appliedto the case of random V and to that of isolated eigenvalues (for any V ).Using the notations of section 3, for j ∈ { , · · · , L } , we define(3.17) S L,j ( E ) := X k = j a N k λ k − E and Γ L,j ( E ) := X k = j λ k − E (cid:18) | ϕ k ( L ) | ϕ k (0) ϕ k ( L ) ϕ k (0) ϕ k ( L ) | ϕ k (0) | (cid:19) . We prove
Theorem 3.3.
Pick j ∈ { , · · · , L } such that − < λ j − + λ j < λ j +1 + λ j < . There exists C > (depending only on ( λ j − + λ j ) + 4 and − ( λ j +1 + λ j ) ) such that, for any L , if a j ≤ d j /C ,equation (2.4) and (2.8) has exactly one solution in the set (3.18) R j := ( E ∈ C ; Re E ∈ λ j + Ca j [ − , − Ca j ≤ Im E ≤ − a j d j /C ) . Moreover, the solution to (2.4) , say z N j , satisfies (3.19) z N j = λ j + a N j S L,j ( λ j ) + e − iθ ( λ j ) + O (cid:18)(cid:16) a N j d − j (cid:17) (cid:19) . and the solution to (2.8) , say z Z j , satisfies (3.20) z Z j = λ j + (cid:28)(cid:18) ϕ j ( L ) ϕ j (0) (cid:19) , (cid:16) Γ L,j ( λ j ) + e − iθ ( λ j ) (cid:17) − (cid:18) ϕ j ( L ) ϕ j (0) (cid:19)(cid:29) + O (cid:18)(cid:16) a Z j d − j (cid:17) (cid:19) . Note that, if a N j d − j is small, formula (3.19) gives the asymptotic of the width of the solution z N j ,namely,(3.21) Im z N j = a N j · sin θ ( λ j )[ S L,j ( λ j ) + cos θ ( λ j )] + sin θ ( λ j ) (1 + o (1)) . Recall that sin θ ( λ j ) < H Z L , using the bounds (3.28) and (3.29), we seethat the asymptotic of the imaginary part of the solution z Z j satisfies(3.22) − C a Z j ≤ Im z Z j ≤ − Ca Z j d j . This and (3.21) will be useful when a j ≪ d j as will be the case for random potentials. The casewhen a j and d j are of the same order of magnitude requires more information. This is the casethat we meet in the next section when dealing with periodic potentials.The proof of Theorem 3.3 also yields the behavior of the functions E S L ( E ) + e − iθ ( E ) and E det (cid:16) Γ L ( E ) + e − iθ ( E ) (cid:17) near their zeros in R j and, in particular shows the following Proposition 3.1.
Fix δ > . Under the assumptions of Theorem 3.3, there exists c > such that,for − δ < λ j − + λ j < λ j +1 + λ j < − δ , one has inf Proposition 3.1 is a consequence of the analogues of (3.24) and (3.30) on the rectangles˜ R j = ˜ z j + ca • j d − j [ − , × [ − , • ∈ { N , Z } and c sufficiently small. Proof of Theorem 3.3. Let us start with equation (2.4). To prove the statement on equation (2.4),in R j , we compare the function E S L ( E ) + e − iθ ( E ) to the function E ˜ S L,j ( E ) = a N j λ j − E + S L,j ( λ j ) + e − iθ ( λ j ) . Clearly, in C , the equation ˜ S L,j ( E ) = 0 admits a unique solution given by˜ z j = λ j + a N j S L,j ( λ j ) + e − iθ ( λ j ) . For E ∈ ∂R j , the boundary of R j , one has (cid:12)(cid:12)(cid:12) ˜ S L,j ( E ) (cid:12)(cid:12)(cid:12) ≥ C and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a N j λ j − E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C , (cid:12)(cid:12)(cid:12) e − iθ ( E ) − e − iθ ( λ j ) (cid:12)(cid:12)(cid:12) ≤ Ca N j and | S L,j ( E ) − S L,j ( λ j ) | ≤ Ca N j d − j . (3.23)Hence, as d j ≤ 1, one gets max E ∈ ∂R j (cid:12)(cid:12)(cid:12) ˜ S L,j ( E ) − S L ( E ) − e − iθ ( E ) (cid:12)(cid:12)(cid:12) | ˜ S L,j ( E ) | ≤ Ca N j d − j Thus, by Rouch´e’s theorem, equation (2.4) has a unique solution in R j .To obtain the asymptotics of the solution, it suffices to use Rouch´e’s theorem again with thefunctions ˜ S L,j and S L ( E ) + e − iθ ( E ) on the smaller rectangle ˜ R j = ˜ z j + K ( a N j d − j ) [ − , × [ − , E ∈ ∂ ˜ R j (cid:12)(cid:12)(cid:12) ˜ S L,j ( E ) − S L ( E ) − e − iθ ( E ) (cid:12)(cid:12)(cid:12) | ˜ S L,j ( E ) | ≤ CK − . Thus, for K sufficiently large, this completes the proof of the statements on the solutions toequation (2.4) contained in Theorem 3.3.Let us turn to equation (2.8). On R j , we now compare Γ L ( E ) + e − iθ ( E ) to the matrix valuedfunction E ˜Γ L,j ( E ) := 1 λ j − E (cid:18) | ϕ j ( L ) | ϕ j (0) ϕ j ( L ) ϕ j (0) ϕ j ( L ) | ϕ j (0) | (cid:19) + Γ L,j ( λ j ) + e − iθ ( λ j ) . The matrix (cid:18) | ϕ j ( L ) | ϕ j (0) ϕ j ( L ) ϕ j (0) ϕ j ( L ) | ϕ j (0) | (cid:19) is rank 1 and can be diagonalized as (cid:18) | ϕ j ( L ) | ϕ j (0) ϕ j ( L ) ϕ j (0) ϕ j ( L ) | ϕ j (0) | (cid:19) = P j (cid:18) a Z j 00 0 (cid:19) P ∗ j where a Z j is given by (2.13) and P j = 1 q a Z j (cid:18) ϕ j ( L ) − ϕ j (0) ϕ j (0) ϕ j ( L ) (cid:19) . Thus, ˜Γ L,j ( E ) is unitarily equivalent to(3.25) M := 1 λ j − E (cid:18) a Z j 00 0 (cid:19) + P ∗ j Γ L,j ( λ j ) P j + e − iθ ( λ j ) . As P ∗ j Γ L,j ( λ j ) P j is real and the imaginary part of e − iθ ( λ j ) does not vanish, the matrix M := P ∗ j Γ L,j ( λ j ) P j + e − iθ ( λ j ) is invertible. By rank 1 perturbation theory (see , e.g., [37]), we know that M is invertible if and only if a Z j (cid:2) M − (cid:3) + λ j = E (where [ M ] is the upper right coefficient ofthe 2 × M ). In this case, one has(3.26) M − = M − − a Z j a Z j (cid:2) M − (cid:3) + λ j − E M − (cid:18) (cid:19) M − . Hence, 0 is an eigenvalue of M if and only if E = λ j + a Z j (cid:20)(cid:16) P ∗ j Γ L,j ( λ j ) P j + e − iθ ( λ j ) (cid:17) − (cid:21) = λ j + (cid:28)(cid:18) ϕ j ( L ) ϕ j (0) (cid:19) , (cid:16) Γ L,j ( λ j ) + e − iθ ( λ j ) (cid:17) − (cid:18) ϕ j ( L ) ϕ j (0) (cid:19)(cid:29) . (3.27)Note that, as Γ L,j ( λ j ) is real symmetric and k Γ L,j ( λ j ) k ≤ Cd − j , one has(3.28) (cid:12)(cid:12)(cid:12)(cid:12)(cid:28)(cid:18) ϕ j ( L ) ϕ j (0) (cid:19) , (cid:16) Γ L,j ( λ j ) + e − iθ ( λ j ) (cid:17) − (cid:18) ϕ j ( L ) ϕ j (0) (cid:19)(cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ a Z j | sin θ ( λ j ) | . and(3.29) Im (cid:18)(cid:28)(cid:18) ϕ j ( L ) ϕ j (0) (cid:19) , (cid:16) Γ L,j ( λ j ) + e − iθ ( λ j ) (cid:17) − (cid:18) ϕ j ( L ) ϕ j (0) (cid:19)(cid:29)(cid:19) ≤ a Z j d j sin θ ( λ j )1 + d j . Using (3.25), (3.26), (3.28) and (3.29),we see that, for E ∈ ∂R j , the boundary of R j , ˜Γ L,j ( E ) isinvertible and that one has (cid:13)(cid:13)(cid:13)(cid:13)h ˜Γ L,j ( E ) i − (cid:13)(cid:13)(cid:13)(cid:13) ≤ C and k Γ L,j ( E ) − Γ L,j ( λ j ) k ≤ Ca Z j d − j . Hence, as d j ≤ 1, taking (3.23) into account, one getsmax E ∈ ∂R j (cid:13)(cid:13)(cid:13)(cid:13) − h ˜Γ L,j ( E ) i − (cid:16) Γ L ( E ) + e − iθ ( E ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C a Z j d − j In the same way, one proves(3.30) max E ∈ ∂ ˜ R j (cid:13)(cid:13)(cid:13)(cid:13) − h ˜Γ L,j ( E ) i − (cid:16) Γ L ( E ) + e − iθ ( E ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) . K − where we recall that ˜ R j = ˜ z j + K ( a N j d − j ) [ − , × [ − , ∂R j and ∂ ˜ R j (for K sufficiently large), det (cid:16) ˜Γ L,j ( E ) (cid:17) and det (cid:16) Γ L ( E ) + e − iθ ( E ) (cid:17) ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 31 as (cid:12)(cid:12)(cid:12) det (cid:16) ˜Γ L,j ( E ) (cid:17) − det (cid:0) Γ L ( E ) + e − iθ ( E ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) ˜Γ L,j ( E ) (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − det (cid:18) − (cid:20) − h ˜Γ L,j ( E ) i − (cid:16) Γ L ( E ) + e − iθ ( E ) (cid:17)(cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . We then conclude as in the case of equation (2.4). This completes the proof of Theorem 3.3. (cid:3) Combining Theorems 3.3, 3.1 and 3.2, we get a pretty clear picture of the resonances near theDirichlet eigenvalues in ( − , 2) as long as the associated a j and d j behave correctly. As said, thisand the knowledge of the spectral statistics for random operators will enable us to prove the resultsdescribed in section 1.3. For the periodic case, Theorems 3.1, 3.2 and 3.3 will prove not too besufficient. As we shall see, in this case, a j and d j are of the same order of magnitude. Thus,neighboring Dirichlet eigenvalues have a sizable effect on the location of resonances. Therefore, inthe next section, we compute the Dirichlet spectral data for the truncated periodic potential.4. The Dirichlet spectral data for periodic potentials As we did not find any suitable reference for this material, we first derive a suitable descriptionof the spectral data (i.e. the ( a j ) j and ( λ j ) j ) for the Dirichlet restriction of a periodic operator tothe interval J , L K when L becomes large.Consider a potential V : N → R such that, for some p ≥ 1, one has V k = V k + p for all k ≥ 0. Weassume p to be minimal, i.e., to be the period of V . In our first result, we describe the spectrum of H Z = − ∆ + V on ℓ ( Z ) and H N = − ∆ + V on ℓ ( N ) (with Dirichlet boundary conditions at 0). Inthe second result we turn to H L , the Dirichlet restriction H N to J , L K and described its spectraldata, i.e., its eigenvalues and eigenfunctions.We recall Theorem 4.1. The spectrum of H Z , say Σ Z , is a union of at most p disjoint intervals that allconsist in purely absolutely continuous spectrum.The spectrum of H N is the union of Σ Z and at most finitely many simple eigenvalues outside Σ Z ,say, ( v j ) ≤ j ≤ n . Σ Z consists of purely absolutely continuous spectrum of H N and the eigenfunctionsassociated to ( v j ) ≤ j ≤ n , say ( ψ j ) ≤ j ≤ n , are exponentially decaying at infinity. Except for the exponential decay of the eigenfunctions, the proof of the statement for the periodicoperator on Z and N is classical and can e.g. be found in a more general setting in [39, chapters 2,3 and 7] (see also [42, 35]). The exponential decay is an immediate consequence of Floquet theoryfor the periodic Hamiltonian on Z and the fact that the eigenvalues lie in gaps of Σ Z .For H Z one can define its Bloch quasi-momentum (see the beginning of section 4.1 for details) thatwe denote by θ p ; it is continuous and strictly increasing on Σ Z and real analytic on ◦ Σ Z . DecomposeΣ Z into its connected components, i.e., Σ Z = q [ r =1 B r where q ≤ p . Let c q be the number of closedgaps contained in q . Then, θ p is continuous and strictly increasing on B r and real analytic on ◦ B r ,the interior of the r -th band. Moreover, on this set, its derivative can be expressed in terms of thedensity of states defined in (1.2) as(4.1) n ( λ ) = 1 π θ ′ p ( λ ) . We first describe the eigenvalues of H L . Theorem 4.2. One has (1) For any k ∈ { , · · · , p − } , there exists h k : Σ Z → R , a continuous function that is realanalytic in a neighborhood of ◦ Σ Z such that, for L sufficiently large s.t. L ≡ k mod p , (a) for ≤ r ≤ q , the function h k maps B r into ( − ( c r + 1) π, ( c r + 1) π ) ; (b) define the function (4.2) θ p,L := θ p − h k L − k ; it is continuous and strictly monotonous on each B r ( ≤ r ≤ q ); (c) for ≤ r ≤ q , the eigenvalues of H L in B r , the r -th band of Σ Z , say ( λ rj ) j , are thesolutions (in Σ Z ) to the quantization conditions (4.3) θ p,L ( λ rj ) = jπL − k , j ∈ Z . (2) There exists c > such that, if λ is an eigenvalue of H L outside Σ Z , then, for L = N p + k sufficiently large, there exists λ ∞ ∈ Σ +0 ∪ Σ − k \ Σ Z s.t., one has | λ − λ ∞ | ≤ e − cL . Recall that Σ +0 and Σ − k are respectively the spectra of H +0 and H − k defined in section 1.2.2.In Theorem 4.2, when solving equation (4.3), one has to do it for each band B r , and, for eachband and each j such that jπL − k ∈ θ p,L ( B r ), equation (4.3) admits a unique solution. But, it mayhappen that one has two solutions to (4.3) for a given j belonging to neighboring bands. In thesequel to simplify the notations, we will not distinguish between the different bands, i.e., we willwrite eigenvalues ( λ j ) j not referring to the band they belong to.Let us now describe the associated eigenfunctions. Theorem 4.3. Recall that ( λ j ) j are the eigenvalues of H L in Σ Z (enumerated as in Theorem 4.2). (1) There exist p + 2 positive functions, say, f +0 , ( f − k ) ≤ k ≤ p − and ˜ f , that are real analyticin a neighborhood of ◦ Σ Z such that, there exists σ r ∈ { +1 , − } such that, for L = N p + k sufficiently large, for λ j in ◦ B r , the interior of r -th band of Σ Z , one has | ϕ l ( L ) | = f − k ( λ j ) L − k f ( λ j ) L − k ! − , | ϕ l (0) | = f +0 ( λ j ) f − k ( λ j ) | ϕ l ( L ) | ,ϕ l ( L ) ϕ l (0) = σ r e iπl | ϕ l ( L ) || ϕ l (0) | = σ r e i ( L − k ) θ p ( λ j ) − h k ( λ j ) | ϕ l ( L ) || ϕ l (0) | . (4.4)(2) Let λ be an eigenvalue of H L outside Σ Z (see point (3) in Theorem 4.2). If ϕ is a normalizedeigenfunction associated to λ and H L , one has one of the following alternatives for L large (a) if λ ∞ ∈ Σ +0 \ Σ − k , one has (4.5) | ϕ ( L ) | ≍ e − cL and | ϕ (0) | ≍ if λ ∞ ∈ Σ − k \ Σ +0 , one has (4.6) | ϕ ( L ) | ≍ and | ϕ (0) | ≍ e − cL ;(c) if λ ∞ ∈ Σ − k ∩ Σ +0 , one has (4.7) | ϕ ( L ) | ≍ and | ϕ (0) | ≍ . For later use, let us define θ p,L , f ,L and f k,L by(4.8) f k,L ( λ ) = f − k ( λ ) f ( λ ) L − k ! − and f ,L ( λ ) = f +0 ( λ ) f ( λ ) L − k ! − where θ p , h k , f , f k and ˜ f are defined in Theorem 4.2.As a consequence of Theorem 4.2, we obtain Corollary 4.1. For λ ∈ ◦ Σ Z , for L ≡ k mod ( p ) sufficiently large, one has dN − k dλ ( λ ) = n − k ( λ ) = f − k ( λ ) n ( λ ) = 1 π f − k ( λ ) θ ′ p ( λ ) = 1 π f k,L ( λ ) θ ′ p,L ( λ ) , (4.9) dN +0 dλ ( λ ) = n +0 ( λ ) = f +0 ( λ ) n ( λ ) = 1 π f +0 ( λ ) θ ′ p ( λ ) = 1 π f ,L ( λ ) θ ′ p,L ( λ ) . (4.10) Here, θ P , f +0 and f − k are defined the functions defined in Theorem 4.2.Proof of Corollary 4.1. To prove the first equalities in (4.9) and (4.10), it suffices to prove that, forany χ ∈ C ∞ ( ◦ Σ Z ), h δ , χ ( H − k ) δ i = Z R χ ( λ ) dN − k ( λ ) = 1 π Z R χ ( θ − p ( k )) f − k ( θ − p ( k )) dk = 1 π Z R χ ( λ ) f − k ( λ ) θ ′ p ( λ ) dλ, (4.11) h δ , χ ( H +0 ) δ i = Z R χ ( λ ) dN +0 ( λ ) = 1 π Z R χ ( θ − p ( k )) f +0 ( θ − p ( k )) dk = 1 π Z R χ ( λ ) f +0 ( λ ) θ ′ p ( λ ) dλ, (4.12)the full statement then following by standard density argument. The operator H L converges to H +0 in norm resolvent sense. Thus, we know that h δ , χ ( H +0 ) δ i = lim L → + ∞ h δ , χ ( H L ) δ i . Now, byTheorem 4.2, as χ is supported in ◦ Σ Z , using the Poisson formula, one computes h δ ,χ ( H L ) δ i = X j χ ( λ j ) || ϕ j (0) | = 1 L − k X l χ (cid:18) θ − p,L (cid:18) lπL − k (cid:19)(cid:19) f ,L (cid:18) θ − p,L (cid:18) lπL − k (cid:19)(cid:19) = 1 L − k X j ∈ Z Z R e − i πjλ χ (cid:18) θ − p,L (cid:18) π λL − k (cid:19)(cid:19) f ,L (cid:18) θ − p,L (cid:18) π λL − k (cid:19)(cid:19) dλ = 1 π X j ∈ Z Z R e − i L − k ) jθ p,L ( λ ) χ ( λ ) f ,L ( λ ) θ ′ p,L ( λ ) dλ. Thus, using the non stationary phase, i.e., integrating by parts, one gets, for any N ≥ (cid:12)(cid:12)(cid:12)(cid:12) h δ , χ ( H L ) δ i − π Z R χ ( λ ) f ,L ( λ ) θ ′ p,L ( λ ) dλ (cid:12)(cid:12)(cid:12)(cid:12) ≤ X j ≥ C N,K k χ k C N ( | j | ( L − k )) − N ≤ C N,K k χ k C N (( L − k )) − N . (4.13)Here, we have used the analyticity of the functions θ p,L and f ,L .To deal with H − k , we recall the operator ˜ H L (that is unitarily equivalent to H L ) defined in Re-mark 1.4. One has h δ L , H L δ L i = h δ , χ ( ˜ H L ) δ i , thus, as H − k is the strong resolvent sense limit of˜ H L , one gets h δ , χ ( H − k ) δ i = lim L → + ∞ h δ L , χ ( H L ) δ L i .Then, (4.11) and (4.12) and, thus, the first equalities in (4.9) and (4.10), follow as θ ′ p,L , f ,L and f k,L converge (locally uniformly on ◦ Σ Z ) respectively to θ ′ p , f +0 and f − k (see (4.8) and Theorem 4.2). Let us now prove the second equalities in (4.9) and (4.10). Therefore, we use an almost analyticextension (see [30]) of χ , say, ˜ χ , that is, a function ˜ χ : C → C satisfying ((1) for z ∈ R , ˜ χ ( z ) = χ ( z ),(2) supp( ˜ χ ) ⊂ { z ∈ C ; | Im( z ) | < } ,(3) ˜ χ ∈ S ( { z ∈ C ; | Im( z ) | < } ),(4) The family of functions x ∂ ˜ χ∂z ( x + iy ) · | y | − n (for 0 < | y | < 1) is bounded in S ( R ) for any n ∈ N .Moreover, ˜ χ can be chosen so that one has the following estimates: for n ≥ α ≥ β ≥ 0, thereexists C n,α,β > < | y |≤ sup x ∈ R (cid:12)(cid:12)(cid:12)(cid:12) x α ∂ β ∂x β (cid:18) | y | − n · ∂ ˜ χ∂z ( x + iy ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C n,α,β sup β ′ ≤ n + β +2 α ′ ≤ α sup x ∈ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x α ′ ∂ β ′ χ∂x β ′ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By the definition of χ , the right hand side of (4.14) is bounded uniformly in E complex.Let χ ∈ C ∞ ( R ) and ˜ χ be an almost analytic extension of χ ( x ). Then, by [15] and [20], we knowthat, for any n and ω ∈ Ω, the following formula hold,(4.15) χ ( H • ) = i π Z C ∂ ˜ χ∂z ( z ) · ( z − H • ) − dz ∧ dz where H • = H L , ˜ H L , H +0 or H − k .Using the geometric resolvent equation (see, e.g., [19, Theorem 5.20]) and the Combes-Thomasestimate (see , e.g., [19, Theorem 11.2]), we know that for some C > 0, for Im z = 0,(4.16) (cid:12)(cid:12)(cid:12)D δ , h ( ˜ H L − z ) − − ( H − k − z ) − i δ E(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:10) δ , (cid:2) ( H L − z ) − − ( H +0 − z ) − (cid:3) δ (cid:11)(cid:12)(cid:12) ≤ C | Im z | e − L | Im z | /C . Plugging (4.16) into (4.15) and using (4.14), we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X j =0 χ ( λ j ) | ϕ j (0) | − Z R χ ( λ ) dN +0 ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˜ C N Z | y |≤ | y | N − e − L | y | /C dy ≤ C N L − N Thus, by (4.12) and (4.13), we obtain that, for χ ∈ C ∞ ( ◦ Σ Z ) and any N ≥ 0, there exists C N > (cid:12)(cid:12)(cid:12)(cid:12)Z R χ ( λ ) (cid:2) f ,L ( λ ) θ ′ p,L ( λ ) − f +0 ( λ ) θ ′ p ( λ ) (cid:3) dλ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R χ ( λ ) f ,L ( λ ) θ ′ p,L ( λ ) dλ − Z R χ ( λ ) dN +0 ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C N L − N . (4.17)Now, by (4.3) and (4.8), the function f ,L θ ′ p,L − f +0 θ ′ p admits an expansion in inverse powers of L that is converging uniformly on compact subsets of ◦ Σ Z , namely, f ,L θ ′ p,L − f +0 θ ′ p = X k ≥ L − k α k . Thus, (4.17) immediately yields that, for any k ≥ 1, one has α k ≡ ◦ Σ Z . Hence, f ,L θ ′ p,L ≡ f +0 θ ′ p on ◦ Σ Z . This completes the proof of Corollary 4.1. (cid:3) ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 35 The proofs of Theorems 4.2 and 4.3. We will first describe some objects from the spectraltheory of H Z , use them to describe the spectral theory of H N , prove Theorem 4.2 and finally proveTheorem 4.3.4.1.1. The spectral theory of H Z . This material is classical (see, e.g., [42, 39]); we only recall it forthe readers convenience. For 0 ≤ j ≤ p − 1, define ˜ T j = ˜ T j ( E ) to be a monodromy matrix for theperiodic finite difference operator H Z , that is ,(4.18) ˜ T j ( E ) = T j + p − ,j ( E ) = T j + p − ( E ) · · · T j ( E ) =: a jp ( E ) b jp ( E ) a jp − ( E ) b jp − ( E ) ! where(4.19) T j ( E ) = (cid:18) E − V j − 11 0 (cid:19) . The coefficients of ˜ T j ( E ) are monic polynomials in the energy E : a jp ( E ) has degree p and b jp ( E )has degree p − 1. Clearly, det ˜ T j ( E ) = 1. As j V j is p -periodic, so is j ˜ T j ( E ). Moreover, for j ′ < j , one has(4.20) ˜ T j ( E ) T j,j ( E ) = T j + p − ,j ′ + p − ( E ) ˜ T j ′ ( E ) = T j,j ′ ( E ) ˜ T j ′ ( E ) . Thus, the discriminant ∆( E ) := tr ˜ T j ( E ) = a jp ( E ) + b jp − ( E ) is a polynomial of degree p thatis independent of j ; so are ρ ( E ) and ρ − ( E ), the eigenvalues of ˜ T j ( E ). One defines the Blochquasi-momentum E θ p ( E ) by(4.21) ∆( E ) = ρ ( E ) + ρ − ( E ) = 2 cos( p θ p ( E )) . Let us recall some basic properties of the discriminant ∆ and the coefficients of ˜ T j , the proofs ofwhich can be found in [42]:(1) if ∆ ′ ( E ) = 0 then | ∆( E ) | ≥ ′ are simple;(3) E is a zero of ∆ ′ s.t. | ∆( E ) | = 2 if and only if ˜ T j ( E ) ∈ { + Id, − Id } (for any j );(4) the polynomials b jp and a jp − only vanish in the set {| ∆( E ) | ≥ } ; they keep a fixed sign ineach of the connected components of the set {| ∆( E ) | < } .Note that ∆( E ) is real when E is real. Thus, for E real, | ∆( E ) | ≤ ρ − ( E ) = ρ ( E ) and | ∆( E ) | > ρ ( E ) is real. When | ∆( E ) | ≤ 2, we will fix ρ ( E ) := e ipθ p ( E ) and when | ∆( E ) | > ρ ( E ) so that | ρ ( E ) | < E belongs to the spectrum of H Z (i.e. − ∆ + V on ℓ ( Z )) if and only if | ∆( E ) | ≤ E a zero of ∆ ′ such that ∆( E ) = ± θ p is real analyticnear E and θ ′ p ( E ) = 0. Definition 4.1. E is said to be a closed gap if and only if | ∆( E ) | = 2 and ∆ ′ ( E ) = 0 orequivalently if and only if ˜ T ( E ) is diagonal.Consider ∂ Σ Z . It is the set of energies solutions to | ∆( E ) | = 2 where ˜ T ( E ) is not diagonal; it isalso the set of roots of | ∆( E ) | = 2 that are not closed gaps. From the upper half of the complexplane, one can continue E θ p ( E ) analytically to the universal cover of C \ ∂ Σ Z . Each of thepoints in ∂ Σ Z is a branch point of θ p of square root type. Moreover, for E ∂ Σ Z , there existstwo linearly independent solutions to the eigenvalue equation ( − ∆ + V − E ) u = 0, say ϕ ± ( E ),satisfying, for n ∈ Z (4.22) ϕ ± ( n + p, E ) = e ± ipθ p ( E ) ϕ ± ( n, E ) . The spectral theory of H N . Let us now turn to the spectrum of the operator on the half-lattice. The operator H +0 . For the operator H +0 = H N (that is − ∆ + V on ℓ ( N ) with Dirichlet boundaryconditions at 0), E is in the spectrum if and only if • either | ∆( E ) | ≤ • or | ∆( E ) | > T ( E )] n (cid:18) (cid:19) stays bounded as n → + ∞ .The second condition is equivalent to asking that [ ˜ T j ( E )] n T j − ( E ) · · · T ( E ) (cid:18) (cid:19) stay bounded as n → + ∞ .When | ∆( E ) | 6 = 2 and a p − ( E ) = 0, one can diagonalize ˜ T ( E ) in the following way(4.23) (cid:18) a p − ( E ) ρ ( E ) − a p ( E ) − a p − ( E ) a p ( E ) − ρ − ( E ) (cid:19) × ˜ T ( E )= (cid:18) ρ ( E ) 00 ρ − ( E ) (cid:19) × (cid:18) a p − ( E ) ρ ( E ) − a p ( E ) − a p − ( E ) a p ( E ) − ρ − ( E ) (cid:19) . Thus, using(4.24) (cid:12)(cid:12)(cid:12)(cid:12) ρ ( E ) − a p ( E ) − b p ( E ) − a p − ( E ) ρ ( E ) − b p − ( E ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ρ ( E ) − a p ( E ) − b p ( E ) − a p − ( E ) a p ( E ) − ρ − ( E ) (cid:12)(cid:12)(cid:12)(cid:12) = 0for n ∈ Z , one computes(4.25) (cid:16) ˜ T ( E ) (cid:17) n = (cid:18) ˜ t ,n ( E ) ˜ t ,n ( E )˜ t ,n ( E ) ˜ t ,n ( E ) (cid:19) where(4.26) ˜ t ,n ( E ) := ρ n ( E ) a p ( E ) − ρ − ( E ) ρ ( E ) − ρ − ( E ) + ρ − n ( E ) ρ ( E ) − a p ( E ) ρ ( E ) − ρ − ( E ) , ˜ t ,n ( E ) := (cid:0) ρ − n ( E ) − ρ n ( E ) (cid:1) b p ( E ) ρ ( E ) − ρ − ( E ) , ˜ t ,n ( E ) := (cid:0) ρ n ( E ) − ρ − n ( E ) (cid:1) a p − ( E ) ρ ( E ) − ρ − ( E ) , ˜ t ,n ( E ) := ρ − n ( E ) a p ( E ) − ρ − ( E ) ρ ( E ) − ρ − ( E ) + ρ n ( E ) ρ ( E ) − a p ( E ) ρ ( E ) − ρ − ( E ) . Clearly, the formulas (4.23), (4.25) and (4.26) stay valid even if a p − ( E ) = 0. They also stay validif | ∆( E ) | = 2 and ∆ ′ ( E ) = 0. Indeed, by points (1)-(3) in section 4.1.1, the functions ρ − ρ − , a p − ρ − , − ρ − a p , b p and a p − are analytic near and have simple zeros at such points.We have thus proved that Lemma 4.1. For E ∂ Σ Z , (cid:16) ˜ T ( E ) (cid:17) n has the form (4.25) - (4.26)Simple computations then show that E is in the spectrum of H +0 , that is, − ∆ + V on ℓ ( N ) withDirichlet boundary conditions at 0 if and only if one of the following conditions is satisfied:(1) | ∆( E ) | ≤ 2: moreover, the set { E ∈ R ; | ∆( E ) | ≤ } is contained in the absolutely contin-uous spectrum of H +0 ;(2) | ∆( E ) | > a p − ( E ) = 0 and | a p ( E ) | < . ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 37 Thus, on Σ Z , the spectrum of H +0 is purely absolutely continuous; it does not contain any embeddedeigenvalues.Note that, in case (2), [ ˜ T ( E )] n (cid:18) (cid:19) actually decays exponentially fast. In this case, E is aneigenvalue associated to the (non normalized) eigenfunction ( u l ) l ∈ N where, for n ≥ j ∈{ , · · · , p − } , u np + j ( E ) = (cid:28) T j − ( E ) · · · T ( E ) (cid:18) (cid:19) , (cid:18) (cid:19)(cid:29) · (cid:2) a p ( E ) (cid:3) n = a j ( E ) (cid:2) a p ( E ) (cid:3) n (4.28)writing(4.29) T j − ( E ) · · · T ( E ) =: (cid:18) a j ( E ) b j ( E ) a j − ( E ) b j − ( E ) (cid:19) . It is well know that, for any j , the zeros of a j and b j are simple (see, e.g., [39, section 4]), andthe roots of a j +1 (resp. b j +1 ) interlace those of a j (resp. b j ). Let E ′ be an eigenvalue of H +0 .Differentiating (4.24) at the energy E ′ , we compute(4.30) b p ( E ′ ) da p − dE ( E ′ ) + ( ρ ( E ′ ) − ρ − ( E ′ )) d ( ρ − a p ) dE ( E ′ ) = 0 . The eigenvalues of the operator H − k . Let us now turn to H − k . Recalling (4.29) and using therepresentation (4.25), we obtain that the eigenvalues of H − k outside Σ Z satisfy(4.31) (cid:18) ρ ( E ) − a p ( E ) − a p − ( E ) − b p ( E ) a p ( E ) − ρ − ( E ) (cid:19) (cid:18) a k +1 ( E ) b k +1 ( E ) (cid:19) = 0 . As for H +0 , the eigenfunction associated to E and H − k decays exponentially fast. Indeed, theeigenvalues of H − k in the region | ∆( E ) | > H +0 , i.e., theyare the energies such that [ ˜ T k ( E )] − n (cid:18) (cid:19) stays bounded; this yields the quantization conditions b kp ( E ) = 0 and | b kp − ( E ) | < 1. In this case, E is an eigenvalue associated to the (non normalized)eigenfunction ( u l ) − l ∈ N where, for n ≥ k ∈ { , · · · , p − } ,(4.32) u − np − k ( E ) = b k ( E ) h b kp − ( E ) i − n . Common eigenvalues to H +0 and H − k . Assume now that E ′ is simultaneously an eigenvalue of H − k and H +0 . In this case, one has a p − ( E ′ ) = 0, | a p ( E ′ ) | < b p ( E ′ ) b k +1 ( E ′ ) = a k +1 ( E ′ )( ρ − ( E ′ ) − ρ ( E ′ )). So (4.31) (see also (4.30)) becomes(4.33) d ( ρ − a p ) dE ( E ′ ) − da p − dE ( E ′ ) − b p ( E ) a p ( E ′ ) − ρ − ( E ′ ) ! (cid:18) a k +1 ( E ′ ) b k +1 ( E ′ ) (cid:19) = 0 . Hence, the analytic function E a k +1 ( E )( a p ( E ) − ρ ( E )) − b k +1 ( E ) a p − ( E ) has a root of order atleast 2 at E ′ . It also implies that a k +1 ( E ′ ) = 0. Indeed, if a k +1 ( E ′ ) = 0, (4.33) implies b k +1 ( E ′ ) = 0as da p − dE ( E ′ ) = 0.Conversely, if E ′ ∈ σ ( H +0 ) such that | ∆( E ′ ) | > E a k +1 ( E )( a p ( E ) − ρ ( E )) − b k +1 ( E ) a p − ( E )has a root of order at least 2 at E ′ , then (4.33) holds and E ′ is an eigenvalue of H − k .We have thus proved Lemma 4.2. E ∈ σ ( H +0 ) ∩ σ ( H − k ) \ Z if and only if | ∆( E ) | > and E is a double root of E a k +1 ( E )( a p ( E ) − ρ ( E )) − b k +1 ( E ) a p − ( E ) . The Dirichlet eigenvalues for a periodic potential : the proof of Theorem 4.2. Let us nowturn to the study of the eigenvalues and eigenvectors of H L , i.e., to the proof of Theorem 4.2. Wefirst prove the statements for the eigenvalues and then, in the next section, turn to the eigenvectors.Recall that L ≡ k mod p ; we write L = N p + k . By definition, E is an eigenvalue of − ∆ + V on J , L K with Dirichlet boundary conditions if and only if0 = det (cid:18) T L +1 ( E ) T L ( E ) T L − ( E ) · · · T ( E ) (cid:18) (cid:19) , (cid:18) (cid:19)(cid:19) = det (cid:18) T k ( E ) · · · T ( E ) · [ ˜ T ( E )] N (cid:18) (cid:19) , (cid:18) (cid:19)(cid:19) (4.34)where ˜ T k ( E ) is the monodromy matrix defined above.We use the notations of sections 4.1.2 and 4.1.1. Let us first show point (1) of Theorem 4.2, namely, Lemma 4.3. For L large, one has ∂ Σ Z ∩ σ ( H L ) = { E ; a k +1 ( E ) = a p − ( E ) = 0 and b p ( E ) = 0 } . Proof. For E ∈ ∂ Σ Z , we know that | ∆( E ) | = 2 and ˜ T ( E ) is not diagonal. Assume ∆( E ) = 2(the case ∆( E ) = − T ( E ) has a Jordan normal form,i.e., there exists P , a 2 × α ∈ R ∗ such that(4.35) ˜ T ( E ) = P − (cid:18) α (cid:19) P where P = (cid:18) p p p p (cid:19) . Thus, by (4.34), E ∈ σ ( H L ) is and only if0 = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) a k +1 ( E ) b k +1 ( E ) a k ( E ) b k ( E ) (cid:19) (cid:16) ˜ T ( E ) (cid:17) N (cid:18) (cid:19) , (cid:18) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) a k +1 ( E ) b k +1 ( E ) a k ( E ) b k ( E ) (cid:19) P − (cid:18) N α (cid:19) P (cid:18) (cid:19) , (cid:18) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (4.36)that is, 0 = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) N α (cid:19) P (cid:18) (cid:19) , P (cid:18) − b k +1 ( E ) a k +1 ( E ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (det P ) a k +1 ( E ) − N α p ( − p b k +1 ( E ) + p a k +1 ( E )) . For N large, this expression vanishes if and only if (det P ) a k +1 ( E ) = 0 and α p ( − p b k +1 ( E ) + p a k +1 ( E )) = 0. As P is invertible, as | b k +1 ( E ) | + | a k +1 ( E ) | 6 = 0 and as α = 0, one has a k +1 ( E ) = 0 and p = 0.In this case, using b k +1 ( E ) = 0, we can then rewrite the eigenvalue equation (4.36) as(4.37) 0 = (cid:12)(cid:12)(cid:12)(cid:12) ( ˜ T ( E )) N (cid:18) (cid:19) , (cid:18) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = ˜ t ,N ( E )For E ∈ ◦ Σ Z close to E , by (4.26), we have t ,N ( E ) = (cid:0) ρ N ( E ) − ρ − N ( E ) (cid:1) a p − ( E ) ρ ( E ) − ρ − ( E ) = ρ N − N − X j =0 ρ − j ( E ) a p − ( E ) . As ρ is continuous at E and ρ ( E ) = 1, taking E to E , we get a p − ( E ) = 0 . As ˜ T ( E ) is not diagonal, this implies b p ( E ) = 0. This completes the proof of Lemma 4.3. (cid:3) ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 39 Now, pick E ∂ Σ Z . Then, by Lemma 4.1, the quantization condition (4.34) becomes(4.38) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ N ( E ) a p ( E ) − ρ − ( E ) ρ ( E ) − ρ − ( E ) + ρ − N ( E ) ρ ( E ) − a p ( E ) ρ ( E ) − ρ − ( E ) − b k +1 ( E ) (cid:0) ρ N ( E ) − ρ − N ( E ) (cid:1) a p − ( E ) ρ ( E ) − ρ − ( E ) a k +1 ( E ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . The eigenvalues outside of Σ Z . Let us first study the eigenvalues outside Σ Z , i.e., in the region | ∆( E ) | > 2. If, for j ∈ N , we define α j ( E ) := a j ( E ) a p ( E ) − ρ − ( E ) ρ ( E ) − ρ − ( E ) + b j ( E ) a p − ( E ) ρ ( E ) − ρ − ( E )and β j ( E ) := a j ( E ) ρ ( E ) − a p ( E ) ρ ( E ) − ρ − ( E ) − b j ( E ) a p − ( E ) ρ ( E ) − ρ − ( E ) , (4.39)equation (4.38) can be rewritten as β k +1 ( E ) + ρ N ( E ) α k +1 ( E ) = 0; using(4.40) α k +1 ( E ) + β k +1 ( E ) = a k +1 ( E ) , (4.38) becomes(4.41) β k +1 ( E ) = − ρ N ( E )1 − ρ N ( E ) a k +1 ( E ) . We first show Lemma 4.4. There exists η > such that, for L sufficiently large, σ ( H L ) ∩ [(Σ Z +[ − η, η ]) \ Σ Z ] = ∅ .Proof. Using (4.39), we rewrite (4.41) as(4.42) a k +1 ( E )( ρ ( E ) − a p ( E )) − b k +1 ( E ) a p − ( E ) = ρ N +1 ( E ) 1 − ρ ( E )1 − ρ N ( E ) a k +1 ( E ) . Pick E ∈ ∂ Σ Z . Then, by our choice for ρ , for η > E ∈ ([ E − η, E + η ]) \ Σ Z , ρ ( E ) = e − c √ | E − E | (1+ O ( √ | E − E | )) . Hence, for E ∈ ([ E − η, E + η ]) \ Σ Z , one has(4.43) (cid:12)(cid:12)(cid:12)(cid:12) ρ N +1 ( E ) 1 − ρ ( E )1 − ρ N ( E ) (cid:12)(cid:12)(cid:12)(cid:12) . min (cid:18)p | E − E | , N (cid:19) . Thus, if a k +1 ( E )( ρ ( E ) − a p ( E )) − b k +1 ( E ) a p − ( E ) = 0, equation (4.42) has no solution in[ E − η, E + η ] \ Σ Z for η small and L sufficiently large.Let us now assume that a k +1 ( E )( ρ ( E ) − a p ( E )) − b k +1 ( E ) a p − ( E ) = 0. Hence, • if a k +1 ( E ) = 0: one computes a k +1 ( E )( ρ ( E ) − a p ( E )) − b k +1 ( E ) a p − ( E ) = a k +1 ( E )( ρ ( E ) − ρ ( E ))(1 + o (1))and ρ N +1 ( E ) 1 − ρ ( E )1 − ρ N ( E ) a k +1 ( E ) = − ( ρ ( E ) − ρ ( E )) a k +1 ( E ) ρ N +1) ( E )1 − ρ N ( E ) (1 + o (1)) . Hence, for η > E ∈ [ E − η, E + η ] \ Σ Z , the two sides of equation (4.42) haveopposite signs: there is no solution to equation (4.42) in this interval; • if a k +1 ( E ) = 0: then b k +1 ( E ) = 0, a p − ( E ) = 0, ρ ( E ) = a p ( E ) and ( a p − ) ′ ( E ) = 0;one computes a k +1 ( E )( ρ ( E ) − a p ( E )) − b k +1 ( E ) a p − ( E ) = − b k +1 ( E )( a p − ) ′ ( E )( E − E )(1 + o (1)) and, by (4.43), for η > E ∈ [ E − η, E + η ] \ Σ Z , (cid:12)(cid:12)(cid:12)(cid:12) ρ N +1 ( E ) 1 − ρ ( E )1 − ρ N ( E ) a k +1 ( E ) (cid:12)(cid:12)(cid:12)(cid:12) . | E − E | min (cid:18)p | E − E | , N (cid:19) Hence, for η > E ∈ [ E − η, E + η ] \ Σ Z , there is no solution to equation (4.42)in this interval.This completes the proof of Lemma 4.4. (cid:3) In Lemma 4.3, we saw that, if E ∈ ∂ Σ Z satisfies a k +1 ( E ) = 0 and a k +1 ( E )( ρ ( E ) − a p ( E )) − b k +1 ( E ) a p − ( E ) = 0, then E is an eigenvalue of H L for L large.By Lemma 4.4, if now suffices to consider energies such that | ∆( E ) | > η for some η > 0. In thiscase, we note that the left hand side in (4.41) is the left hand side of the first equation in (4.31)(up to the factor ρ − ρ − that does not vanish outside Σ Z ). On the other hand, the right hand sidein (4.41) is uniformly exponentially small for large N on { E ∈ R ; | ∆( E ) | > η } . Thus, for L large, the solutions to (4.41) are exponentially close to E ′ that is either an eigenvalue of H +0 or oneof H − k . One distinguishes between the following cases:(1) if E ′ is an eigenvalue of H +0 but not of H − k , then E ′ is a simple root of the function E β k +1 ( E ) (see section 4.1.2); one has to distinguish two cases depending on whether a k +1 ( E ′ ) vanishes or not. Assume first a k +1 ( E ′ ) = 0; then, by (4.28), we know that theeigenvector of H +0 actually satisfies the Dirichlet boundary conditions at L ; thus, E ′ is asolution to (4.41), i.e., an eigenvalue of H L , and (4.28) gives a (non normalized) eigenvector.Assume now that a k +1 ( E ′ ) = 0; then, by Rouch´e’s Theorem, the unique solution to (4.41)close to E ′ satisfies(4.44) E − E ′ = − ρ N ( E ′ ) β ′ k +1 ( E ′ ) a k +1 ( E ′ )(1 + o ( ρ N ( E ′ )));(2) if E ′ is an eigenvalue of H − k but not of H +0 , mutandi mutandis, the analysis is the same asin point (1);(3) if E ′ is an eigenvalue of both H +0 and H − k , then, we are in a resonant tunneling situation.The analysis done in the appendix, section 7, shows that near E ′ , H L has two eigenvalues,say E ± satisfying, for some constant α > E ± − E ′ = ± α ρ N ( E ′ )) (cid:0) O (cid:0) N ρ ( E ′ ) N (cid:1)(cid:1) . This completes the proof of the statements of Theorem 4.2 for the eigenvalues outside Σ Z . The eigenvalues inside Σ Z . We now study the eigenvalues in the region ◦ Σ Z . One can express ρ ( E )in terms of the Bloch quasi-momentum θ p ( E ) and use ρ − ( E ) = ρ ( E ). Notice that, on ◦ Σ Z , one has • Im ρ ( E ) does not vanish • the function E ρ ( E ) is real analytic, • the functions E a p ( E ), E a p − ( E ), E a k +1 ( E ) and E b k +1 ( E ) are real valuedpolynomials.We prove Lemma 4.5. The function α k +1 is analytic and does not vanish on ◦ Σ Z .Proof. Assume that the function α k +1 vanishes at a point E in ◦ Σ Z : • if ρ ( E ) = ρ − ( E ): then, one has a k +1 ( E ) ( a p ( E ) − ρ − ( E )) + b k +1 ( E ) a p − ( E ) = 0: as ρ ( E ) = ρ − ( E ) and E ∈ ◦ Σ Z , one has ρ − ( E ) = ρ ( E ) R ; thus, for a k +1 ( E ) ( a p ( E ) − ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 41 ρ − ( E )) − b k +1 ( E ) a p − ( E ) to vanish, one needs a k +1 ( E ) = 0 and a p − ( E ) = 0 (as b k +1 and a k +1 don’t vanish together); this implies that ρ ( E ) = ± ρ ( E ) = ρ − ( E ); • if ρ ( E ) = ρ − ( E ): such a point E is a simple root of the three functions a p − , ρ − ρ − and a p − ρ that are analytic near E (see points (1)-(4) in section 4.1.1). Moreover, one checksthat the derivatives of these functions at that point are respectively real, purely imaginaryand neither real, nor purely imaginary: for E close to E , one has a p − ( E ) = A ( E − E )(1 + O ( E − E )) ,ρ ( E ) − ρ − ( E ) = 2 iC ( E − E )(1 + O ( E − E )) ,a p ( E ) − ρ − ( E ) = ( B + iC )( E − E )(1 + O ( E − E )) where ( A, B, C ) ∈ ( R ∗ ) . (4.46) Now, as a k +1 and b k +1 are real valued and can’t vanish at the same point, we see that α k +1 ( E ) = 0.This complete the proof of Lemma 4.5 (cid:3) Now, as L = N p + k , the characteristic equation (4.38) (valid for E ∈ ◦ Σ Z ) becomes ρ N ( E ) = e iNpθ p ( E ) = − α k +1 ( E ) α k +1 ( E ) = − β k +1 ( E ) β k +1 ( E )= a k +1 ( E )( ρ ( E ) − a p ( E )) − b k +1 ( E ) a p − ( E ) a k +1 ( E )( ρ ( E ) − a p ( E )) − b k +1 ( E ) a p − ( E ) =: e ih k ( E ) . (4.47)By Lemma 4.5, the function E h k ( E ) defined in (4.47) is real analytic on ◦ Σ Z . Clearly, as insideΣ Z , ρ is real only at bands edges or closed gaps, h k takes values in π Z only at bands edges or closedgaps. This implies point (a) of Theorem 4.2. We prove Lemma 4.6. The function h k can be extended continuously from ◦ Σ Z to Σ Z ; for E ∈ ∂ Σ Z , one has h k ( E ) ∈ ( π + π Z if a k +1 ( E ) = 0 and a k +1 ( E )( ρ ( E ) − a p ( E )) − b k +1 ( E ) a p − ( E ) = 0 ,π Z if not . The function θ p,L is strictly increasing on the bands of Σ Z .Proof. Pick E ∈ ∂ Σ Z . It suffices to study the behavior of E ∈ Σ Z s ( E ) := a k +1 ( E )( ρ ( E ) − a p ( E )) − b k +1 ( E ) a p − ( E ) near E inside Σ Z . Write E = E ± t for t real positive; here, the sign ± depends on whether E is a left or right edge of Σ Z and is chosen so that E = E ± t ∈ ◦ Σ Z for t small.First, t ρ ( E ± t ) is analytic near 0; thus, so is t s ( E ± t ). Solving the characteristicequation ρ ( E ) − ∆( E ) ρ ( E ) + 1 = 0, one finds ρ ( E ± t ) = ρ ( E ) + iat + bt + O ( t ) , a ∈ R ∗ , b ∈ R . Thus, s ( E ± t ) = s ( E ) + ia k +1 ( E ) · a · t + c · t + O ( t )where c := a ′ k +1 ( E )( ρ ( E ) − a p ( E ))+ a k +1 ( E )( b − ( a p ) ′ ( E )) − ( b ′ k +1 ( E ) a p − ( E )+ b k +1 ( E )( a p − ) ′ ( E )) . Hence, • if s ( E ) = 0, then s ( E ± t ) = s ( E ) + O ( t ) ; hence, h k ( E ± t ) = πn + O ( t ) for some n ∈ Z • if s ( E ) = 0 and a k +1 ( E ) = 0, one has s ( E ± t ) = ia k +1 ( E ) · a · t + O ( t ); thus, h k ( E ± t ) = π + πn + O ( t ) for some n ∈ Z ; • if s ( E ) = a k +1 ( E ) = 0, one has b k +1 ( E ) = 0, a p − ( E ) = 0, ρ ( E ) = a p ( E ) and( a p − ) ′ ( E ) = 0; thus s ( E ± t ) = − b k +1 ( E )( a p − ) ′ ( E ) t + 0( t ); hence, h k ( E ± t ) = πn + O ( t ) for some n ∈ Z .This completes the proof of the statement of Lemma 4.6 on the function h k .Let us now control the monotony of θ p,L (see Theorem 4.2) on the bands of Σ Z . It is well known thatkeeping the above notations, θ p ( E ± t ) − θ p ( E ) = ± αt (1 + tg ( t )) with α > . The computationsdone in the previous paragraph show that h k ( E ± t ) = h k ( E ) + at k (1 + tg ( t )), k ≥ 1. Hence, • if k > 1, we have θ p,L ( E ± t ) − θ p,L ( E ) = ± αt (1 + tg ( t )), • if k = 1, we have θ p,L ( E ± t ) − θ p,L ( E ) = (cid:18) ± α + aL − k (cid:19) t (1 + tg ( t )).Hence, θ p,L is strictly increasing inside the band near E for L sufficiently large. Outside a neighbor-hood of the edges of a band, by analyticity of h k , as the bands are compact, we have | θ ′ p,L − θ ′ p | . L − .As θ p is strictly increasing on each band, θ p,L is also strictly increasing outside a neighborhood ofthe edges of a band. This completes the proof of Lemma 4.6. (cid:3) One proves Lemma 4.7. Let E be a closed gap for H Z (see Definition 4.1). Then, for any L = N p + k thefollowing assertions are equivalent: (4.48) E ∈ σ ( H L ) ⇐⇒ h k ( E ) ∈ π Z ⇐⇒ a k +1 ( E ) = 0 ⇐⇒ α k +1 ( E ) ∈ i R ∗ . Proof. The proof of the first equivalence follows immediately from Definition 4.1 and the quantiza-tion condition (4.47); the second follows from (4.39) and the expansions in (4.46); the third followsLemma 4.6, (4.39) and (4.47). (cid:3) Let us note that, in particular, closed gaps where a k +1 vanishes are eigenvalues of H L for all L = N p + k . Remark 4.1. The characteristic equation (4.47) and the computations done at the end of theproof of Lemma 4.5 show that, for L = N p + k large, an energy E such that ρ ( E ) = ρ − ( E ) isan eigenvalues of H L if and only if a k +1 ( E ) = 0. This is an extension of Lemma 4.3.In view of the definition and monotony of θ p,L , the quantization condition (4.47) is clearly equivalentto (4.3). This completes the proof Theorem 4.1 on the eigenvalues of H L . Let us now turn to thecomputation of the associated eigenfunctions.4.1.4. The Dirichlet eigenfunctions for a truncated periodic potential: the proof of Theorem 4.3. Recall that we assume L = N p + k . First, if ( u jl ) Ll =0 is an eigenfunction associated to the eigenvalue λ j , the eigenvalue equation reads u jl +1 u jl ! = T l ( λ j ) u jl u jl − ! for 0 ≤ l ≤ L where u jL +1 = u j − = 0 . To normalize the solution, we assume that u j = 1. The coefficients we want to compute are(4.49) | ϕ j ( L ) | = | u jL | L X l =0 (cid:12)(cid:12)(cid:12) u jl (cid:12)(cid:12)(cid:12) ! − and | ϕ j (0) | = L X l =0 (cid:12)(cid:12)(cid:12) u jl (cid:12)(cid:12)(cid:12) ! − . ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 43 Fix l = np + m . Thus, using the notations of section 4.1.3 and the expressions (4.25), (4.26)and (4.23), one computes(4.50) u jl u jl − ! = T m − , ( λ j ) (cid:16) ˜ T ( λ j ) (cid:17) n (cid:18) (cid:19) = (cid:18) α m ( λ j ) ρ n ( λ j ) + β m ( λ j ) ρ − n ( λ j ) α m − ( λ j ) ρ n ( λ j ) + β m − ( λ j ) ρ − n ( λ j ) (cid:19) where α m and β m are defined in (4.39). The eigenvectors associated to eigenvalues inside Σ Z . As ρ − ( λ j ) = ρ ( λ j ), β m ( λ j ) = α m ( λ j ) andas the functions ( α m ) ≤ m ≤ p − do not vanish on ◦ Σ Z , we compute(4.51) (cid:12)(cid:12)(cid:12) u jnp + m (cid:12)(cid:12)(cid:12) = 2 | α m ( λ j ) | " α m ( λ j ) α m ( λ j ) ρ n ( λ j ) . As L = N p + k , using the quantization condition (4.47), we obtain that L X l =0 (cid:12)(cid:12)(cid:12) u jl (cid:12)(cid:12)(cid:12) = 2 k X m =0 | α m ( λ j ) | " α m ( λ j ) α m ( λ j ) ρ N ( λ j ) + 2 p − X m =0 | α m ( λ j ) | N − X n =0 " α m ( λ j ) α m ( λ j ) ρ n ( λ j ) = N p f ( λ j ) (cid:18) N p ˜ f ( λ j ) (cid:19) (4.52)where we have defined(4.53) f ( E ) := 2 p p − X m =0 | α m ( E ) | . and, using the quantization condition (4.47), computed˜ f ( E ) := 2 f ( E ) Re " p − X m =0 α m ( E ) ! − ρ ( E ) α k +1 ( E ) α k +1 ( E ) ! + 2 f ( E ) k X m =0 | α m ( E ) | − Re " α m ( E ) α k +1 ( E ) α m ( E ) α k +1 ( E ) (4.54)The function E f ( E ) is real analytic and does not vanish on ◦ Σ Z .We prove Proposition 4.1. For E , a closed gap, one has p − X m =0 α m ( E ) = 0 .Proof. By the definition of ( a j , b j ), see (4.29), and that of α j ( E ), see (4.39), the sequence ( α j ( E )) j ∈ Z satisfies the equation α j +1 + α j − + ( V j − E ) α j = 0. As ˜ T ( E ) = T p − ( E ) · · · T ( E ), by (4.23),for j ∈ Z , one has α j + p ( E ) = ρ ( E ) α j ( E ). Hence, the column vector A ( E ) = ( α ( E ) , · · · , α p ( E )) t satisfies ( H ρ − E ) A ( E ) = 0 where H ρ = V · · · ρ ( E )1 V · · · 00 1 V · · · · · · V p − ρ − ( E ) 0 · · · V p . Thus, we have(4.55) h ( H ρ − E ) A ( E ) , A ( E ) i R = 0where h· , ·i R denotes the real scalar product over C p , i.e., * z ... z p , z ′ ... z ′ p + R = p X j =1 z j z ′ j .The functions E A ( E ) and E ρ ( E ) being analytic over ◦ Σ Z , one can differentiate (4.55) withrespect to E to obtain(4.56) 0 = −h A ( E ) , A ( E ) i R + ( ρ ( E ) − ρ − ( E )) (cid:0) ρ − ( E ) ρ ′ ( E ) α ( E ) α p ( E ) − α p ( E ) α ′ ( E ) + α ( E ) α ′ p ( E ) (cid:1) . Here, we have used the fact that, if H tρ is the transposed of the matrix H ρ , then H tρ − H ρ = ( ρ ( E ) − ρ − ( E )) · · · − · · · · · · 01 0 · · · . At E , a closed gap, one has ρ ( E ) = ρ − ( E ). Hence, (4.56) implies0 = h A ( E ) , A ( E ) i R = p − X m =0 α m ( E ) . This completes the proof of Proposition 4.1. (cid:3) In view of (4.54), the function ˜ f is real analytic on ◦ Σ Z ; indeed, the only poles of the function E [ ρ ( E ) − ρ − ( E )] − in ◦ Σ Z are the closed gaps; they are simple poles of this function and, byProposition 4.1, the real analytic function E p − X m =0 α m ( E ) vanishes at these poles.Now that we have computed the normalization constant, let us compute the coefficient u jL definedin (4.49). As L = N p + k , the characteristic equation for λ j , that is, (4.47) reads(4.57) α k +1 ( λ j ) ρ N ( λ j ) = − β k +1 ( λ j ) ρ − N ( λ j ) = − α k +1 ( λ j ) ρ N ( λ j ) . ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 45 Hence, one computes u jL = α k ( λ j ) ρ N ( λ j ) + α k ( λ j ) ρ N ( λ j ) = ρ N ( λ j ) α k ( λ j ) α k +1 ( λ j ) − α k ( λ j ) α k +1 ( λ j ) α k +1 ( λ j )= − ρ N ( λ j ) a p − ( λ j )( ρ ( λ j ) − ρ − ( λ j )) α k +1 ( λ j ) = − e i [ Npθ p ( λ j ) − h k ( λ j )] a p − ( λ j ) (cid:12)(cid:12)(cid:12) a k +1 ( λ j )( a p ( λ j ) − ρ − ( λ j )) + b k +1 ( λ j ) a p − ( λ j ) (cid:12)(cid:12)(cid:12) = − e iπj a p − ( λ j ) (cid:12)(cid:12)(cid:12) a k +1 ( λ j )( a p ( λ j ) − ρ − ( λ j )) + b k +1 ( λ j ) a p − ( λ j ) (cid:12)(cid:12)(cid:12) (4.58)where we have used the quantization condition satisfied by λ j , the last equality in (4.47), and that (cid:12)(cid:12)(cid:12)(cid:12) α k +1 ( λ j ) α k ( λ j ) α k +1 ( λ j ) α k ( λ j ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a p − ( λ j ) ρ ( λ j ) − ρ − ( λ j ) a p ( λ j ) − ρ − ( λ j ) ρ ( λ j ) − ρ − ( λ j ) − a p − ( λ j ) ρ ( λ j ) − ρ − ( λ j ) ρ ( λ j ) − a p ( λ j ) ρ ( λ j ) − ρ − ( λ j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) b k +1 ( λ j ) b k ( λ j ) a k +1 ( λ j ) a k ( λ j ) (cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a p ( λ j ) − ρ − ( λ j ) ρ ( λ j ) − ρ − ( λ j ) − ρ ( λ j ) − a p ( λ j ) ρ ( λ j ) − ρ − ( λ j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) b k ( λ j ) b k +1 ( λ j ) a k ( λ j ) a k +1 ( λ j ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 Lemma 4.8. Define the function ˜ f − k ( E ) by ˜ f − k ( E ) := | a p − ( E ) | | a k +1 ( E )( a p ( E ) − ρ − ( E )) + b k +1 ( E ) a p − ( E ) | ; Then, the function ˜ f − k does not vanish on ◦ Σ Z .Proof. By the definition of α k +1 , one has ˜ f − k ( E ) = | a p − ( E ) | | ρ ( E ) − ρ − ( E )) | | α k +1 ( E ) | . That this ex-pression is well defined and does not vanish on ◦ Σ Z follows from Lemma 4.5 and the computationsmade in the proof thereof. (cid:3) Plugging (4.58) this and (4.51) into (4.49), recalling that u j = 1, outside the bad closed gaps, weobtain (4.4) if, • in addition to (4.53) and (4.54), we set f +0 ( E ) := 1 f ( E ) and f − k ( E ) = f +0 ( E ) · ˜ f − k ( E ), • we remember that the function a p − only changes sign in the gaps of the spectrum Σ Z (seepoint (4) in section 4.1.1) and set σ r to be the sign of − a p − on B r , the r -th band.By (4.49) and (4.51), we obtain (4.4) using Lemma 4.8. This completes the proof of the statementsin Theorem 4.3 on the eigenfunctions of H L associated to eigenvalues in ◦ Σ Z . Remark 4.2. To complete our study let us also see what happens the eigenfunctions near theedges of the spectrum. Pick E ∈ ∂ Σ Z . One then knows that, for E ∈ Σ Z , E close to E , one has(4.59) θ p ( E ) − θ p ( E ) = a p | E − E | (1 + o (1)) (see the proof of Lemma 4.6).Let us rewrite ˜ f (see (4.54)) in the following way˜ f ( E ) = 2 f ( E ) " p − X m =0 | α m ( E ) | cos( h k ( E ) − h m − ( E ) − pθ p ( E )) sin( h k ( E ))sin( pθ p ( E ))+ 2 f ( E ) k X m =0 | α m ( E ) | (1 − cos(2( h k ( E ) − h m − ( E )))) . (4.60)Let us first show Lemma 4.9. For any ≤ m ≤ p − , E | α m ( E ) | p f ( E ) can be extended continuously from ◦ Σ Z to Σ Z .Proof. For p = 1 there is nothing to be done as | α m ( E ) | p f ( E ) ≡ p ≥ 2, we note that, for 0 ≤ m ≤ m + 1 ≤ p − 1, as (cid:12)(cid:12)(cid:12)(cid:12) a m +1 ( E ) b m +1 ( E ) a m ( E ) b m ( E ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 by (4.29),0 = a m +1 ( E )( a p ( E ) − ρ − ( E )) + b m +1 ( E ) a p − ( E )= a m ( E )( a p ( E ) − ρ − ( E )) + b m ( E ) a p − ( E )if and only if a p − ( E ) = 0 (as this implies a p ( E ) − ρ − ( E ) = 0).Let us assume this is the case. As p ≥ 2, we know that p − X j =0 | a j ( E ) | = 0. By (4.46), for atleast one m ∈ { , · · · , p − } , one has a m ( E ) = 0 and α m ( E ) = bc − a m ( E ) + O ( p | E − E | ).Hence, E | α m ( E ) | p f ( E ) can be continued to E setting 2 | α m ( E ) | p f ( E ) = | a m ( E ) | | a ( E ) | + · · · + | a p − ( E ) | .Actually, f ( E ) can be continued at E by setting(4.61) f ( E ) = | a ( E ) | + · · · + | a p − ( E ) | . Let us now assume that a p − ( E ) = 0. We study the behavior of α m near E . Recall (4.39). Then,one has(1) either d m := a m ( E )( a p ( E ) − ρ − ( E )) + b m ( E ) a p − ( E ) = 0: in this case, by (4.46), onehas α m ( E ) = d m c − √ | E − E | (1 + o (1));(2) or d m = a m ( E )( a p ( E ) − ρ − ( E ))+ b m ( E ) a p − ( E ) = 0: in this case, as for some A m ∈ R ∗ and k m ≥ 1, one has a m ( E )( a p ( E ) − ρ − ( E )) + b m ( E ) a p − ( E ) = A m ( E − E ) k m (1 + o (1)) , and, by (4.46), one can continue α m to E by setting α m ( E ) = a m ( E ) / a p − ( E ) = 0, we know that for some m ∈ { , · · · , p − } , we are in case (a). Hence, one has(4.62) f ( E ) = 2 p | E − E | p − X m =0 | a m ( E )( a p ( E ) − ρ − ( E )) + b m ( E ) a p − ( E ) | (1 + o (1))and E | α m ( E ) | p f ( E ) can be continued to E setting 2 | α m ( E ) | p f ( E ) = | d m | | d | + · · · + | d p − | (using thenotation introduced in point (a).This completes the proof of Lemma 4.9. (cid:3) ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 47 By Lemma 4.6, we know that for 1 ≤ k ≤ p and E ∈ ∂ Σ Z , one has 2 h k ( E ) ∈ π Z . Thus, for1 ≤ k ≤ p , 1 ≤ m ≤ p and E ∈ ∂ Σ Z , one has cos( h k ( E ) − h m − ( E ) − pθ p ( E )) sin( h k ( E )) = 0.Using the expansions leading to the proof of Lemma 4.6, one getscos( h k ( E ) − h m − ( E ) − pθ p ( E )) sin( h k ( E )) = c p | E − E | (1 + o (1)) . Recalling (4.59) and the fact that pθ p ( E ) ∈ π Z , Lemma 4.9 implies that ˜ f can be extendedcontinuously up to E . Hence, the expansion (4.52) again yields(4.63) L X l =0 (cid:12)(cid:12)(cid:12) u jl (cid:12)(cid:12)(cid:12) ≍ N pf ( λ j ) . Let us now review the computation (4.58) in this case. We distinguish two cases:(1) if a p − ( E ) = 0: then, (4.58) and the fact that a k +1 ( E ) = 0 (this case was dealt with inpoint (1)), yields that, for | λ j − E | sufficiently small, | u jL | ≍ q | λ j − E | . By (4.61) and (4.63), we obtain(4.64) | ϕ j ( L ) | ≍ | λ j − E | N p and | ϕ j (0) | ≍ N p . (2) if a p − ( E ) = 0: then(a) if d k +1 = 0 (see case (a) in the proof of Lemma 4.9): by (4.62) and (4.63), one has(4.65) | ϕ j (0) | ≍ | λ j − E | N p and | ϕ j ( L ) | ≍ | λ j − E | N p . (b) if d k +1 = 0: by (4.62) and (4.63), one has(4.66) | ϕ j (0) | ≍ | λ j − E | N p and | ϕ j ( L ) | ≍ N p . The eigenvectors associated to eigenvalues outside Σ Z . Let us now turn to the eigenfunctions asso-ciated to eigenvalues H L in the gaps of Σ Z , i.e., in the region { E ; | ∆( E ) | > } . On R \ Σ Z , theeigenvalue E ρ ( E ) is real valued (recall that we pick it so that | ρ ( E ) | < 1) and so are all thefunctions ( α m ) ≤ m ≤ p − and ( β m ) ≤ m ≤ p − (see (4.39)). For 0 ≤ m ≤ p − 1, (4.50) yields(4.67) (cid:12)(cid:12)(cid:12) u jnp + m (cid:12)(cid:12)(cid:12) = α m ( E ) ρ n ( E ) + β m ( E ) ρ − n ( E ) + 2 α m ( E ) β m ( E ) . As when we studied the eigenvalues of H L , let us now distinguish the cases when E is close to aneigenvalue of H +0 or to an eigenvalue of H − k :(1) Pick E ′ an eigenvalue of H +0 but not an eigenvalue of H − k ; then, recall that a p − ( E ′ ) = 0 = a p ( E ′ ) − ρ ( E ′ ). Thus, for 0 ≤ m ≤ p − 1, one has β m ( E ′ ) = 0. Assume E be close to E ′ .As E satisfies (4.44), using (4.41), (4.67) becomes (cid:12)(cid:12)(cid:12) u jnp + m (cid:12)(cid:12)(cid:12) = ρ n ( E ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α m ( E ′ ) − β ′ m ( E ′ ) β ′ k +1 ( E ′ ) a k +1 ( E ′ ) · (cid:2) ρ ( E ′ ) − ρ − ( E ′ ) (cid:3) ρ N − n ) ( E ′ ) + O ( ρ N ( E )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . for 0 ≤ m ≤ p − ≤ n ≤ N − ≤ m ≤ k if n = N .Using (4.40), one computes(4.68) (cid:12)(cid:12)(cid:12) u jnp + m (cid:12)(cid:12)(cid:12) = ρ n ( E ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a m ( E ′ ) − β ′ m ( E ′ ) β ′ k +1 ( E ′ ) a k +1 ( E ′ ) ρ N − n ) ( E ′ ) + O ( ρ N ( E )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . This yields L X l =0 (cid:12)(cid:12)(cid:12) u jl (cid:12)(cid:12)(cid:12) = p − X m =0 N − X n =0 ρ n ( E ′ ) a m ( E ′ ) + O ( N ρ N ( E ))= 11 − ρ ( E ′ ) p − X m =0 a m ( E ′ ) + O ( N ρ N ( E )) . Moreover, by (4.49), (4.67) and (4.39), as a p − ( E ′ ) = 0 = a p ( E ′ ) − ρ ( E ′ ), we obtain | ϕ j ( L ) | = ρ N ( E ′ ) (1 − ρ ( E ′ )) a k +1 ( E ′ ) (cid:2) β ′ k +1 ( E ′ ) (cid:3) p − X m =0 a m ( E ′ ) (cid:12)(cid:12)(cid:12)(cid:12) β ′ k ( E ′ ) a k ( E ′ ) β ′ k +1 ( E ′ ) a k +1 ( E ′ ) (cid:12)(cid:12)(cid:12)(cid:12) + O ( N ρ N ( E ))= γρ N ( E ′ ) + O ( N ρ N ( E )) . where γ := (1 − ρ ( E ′ )) a k +1 ( E ′ ) (cid:2) β ′ k +1 ( E ′ ) (cid:3) p − X m =0 a m ( E ′ ) da p − dE ( E ′ ) ! > . Hence, | ϕ j ( L ) | is exponentially small in L (recall | ρ ( E ) | < E ′ is an eigenvalue of H − k but not of H +0 , then inverting the parts of H − k and H +0 , we seethat | ϕ j ( L ) | is of order 1. A precise asymptotic can be computed but it won’t be needed.(3) if E ′ is an eigenvalue of H +0 and of H − k , the double well analysis done in section 7 showsthat for normalized eigenvectors, say, ϕ , associated to the two eigenvalues of H L close to E ′ , the four coefficients | ϕ , (0) | and | ϕ , ( L ) | are of order 1. Again precise asymptoticscan be computed but won’t be needed.This completes the description of the eigenfunctions given by Theorem 4.3 and completes the proofof this result. (cid:3) Resonances in the periodic case We are now in the state to prove the results stated in section 1.2. Therefore, we first study thefunction E S L ( E ) and E Γ L ( E ) in the complex strip I + i ( −∞ , 0) for I ⊂ ◦ Σ Z .5.1. The matrix Γ L in the periodic case. Using Theorem 4.2, we first prove Theorem 5.1. Fix I ⊂ ◦ Σ Z a compact interval. There exists ε I > and σ I ∈ { +1 , − } such that,for any N ≥ , there exists C N > such that, for L sufficiently large s.t L ≡ k mod ( p ) , one has (5.1) sup Re E ∈ I − ε I < Im E< (cid:12)(cid:12)(cid:12) Γ L ( E ) − Γ eff L ( E ) (cid:12)(cid:12)(cid:12) ≤ C N L − N . ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 49 where (5.2)Γ eff L ( E ) = − θ ′ p ( E )sin u L ( E ) e − iu L ( E ) f − k ( E ) σ I q f − k ( E ) f +0 ( E ) σ I q f − k ( E ) f +0 ( E ) e − iu L ( E ) f +0 ( E ) + Z R dN − k ( λ ) λ − E Z R dN +0 ( λ ) λ − E and u L ( E ) := ( L − k ) θ p,L ( E ) (see (4.2) ), The sign σ I only deepends on the spectral band containing I .Deeper into the lower half-plane, we obtain the following simpler estimate Theorem 5.2. There exists C > such that, for any ε > and for L ≥ sufficiently large s.t. L = N p + k , one has (5.3) sup Re E ∈ I Im E< − ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ L ( E ) − Z R dN − k ( λ ) λ − E Z R dN +0 ( λ ) λ − E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε − e − εL/C . In sections 5.2, the approximations (5.1) and (5.3) theorems will be used to prove Theorems 1.2, 1.3and 1.4.Let us note that, as cot z = i + O (cid:0) e − i Im z (cid:1) , for ε ∈ (0 , ε I ), the asymptotics given by Theorems 5.1and 5.2 coincide in the region { Re E ∈ I, Im E ∈ ( − ε I , − ε ) } : indeed one has,sup Re E ∈ I − ε I < Im E< − ε (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) θ ′ p ( E )sin u L ( E ) e − iu L ( E ) f − k ( E ) σ I q f − k ( E ) f +0 ( E ) σ I q f − k ( E ) f +0 ( E ) e − iu L ( E ) f +0 ( E ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ e − εL/C . Let us now turn to the proofs of Theorems 5.1 and 5.2.5.1.1. The proof of Theorem 5.1. To prove Theorem 5.1, we split the sum S L ( E ) into two parts,one containing the Dirichlet eigenvalues “close” to Re E , the second one containing those “far”from Re E . By “far”, we mean that the distance to Re E is lower bounded by a small constantindependent of L . The “close” eigenvalues are then described by Theorem 4.2. For the “far”eigenvalues, the strong resolvent convergence of H L to H +0 , that of ˜ H L to H − k (see Remark 1.4)and Combes-Thomas estimates enable us to compute the limit and to show that the prelimit andthe limit are O ( L −∞ ) close to each other. For the “close” eigenvalues, the sum coming up in (2.9),the definition of Γ L , is a Riemann sum. We use the Poisson summation formula to obtain a preciseapproximation.As I is a compact interval in ◦ Σ Z , we pick ε > E ∈ I , one has [ E − ε, E + 6 ε ] ⊂ ◦ Σ Z .Let χ ∈ C ∞ ( R ) be a non-negative cut-off function such that χ ≡ − ε, ε ] and χ ≡ − ε, ε ]. For E ∈ I , define χ E ( · ) = χ ( · − E ).We first give the asymptotic for the sum over the Dirichlet eigenvalues far from Re E . We prove Lemma 5.1. For any N > , there exists C N > such that, for L sufficiently large such that L ≡ k mod ( p ) , one has (5.4) sup E ∈ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X j =1 − χ Re E ( λ j ) λ j − E (cid:18) | ϕ j ( L ) | ϕ j (0) ϕ j ( L ) ϕ j (0) ϕ j ( L ) | ϕ j (0) | (cid:19) − ˜ M ( E ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N L − N where (5.5) ˜ M ( E ) := Z R (1 − χ Re E )( λ ) dN − k ( λ ) λ − E Z R (1 − χ Re E )( λ ) dN +0 ( λ ) λ − E . Proof of Lemma 5.1. Recall (see Theorem 2.1) that H L is the operator H +0 restricted to J , L K withDirichlet boundary condition at L ; as L ≡ k mod ( p ), it is unitarily equivalent to the operator H − k restricted to J − L, K with Dirichlet boundary condition at − L (see Remark 1.4).Pick ˜ χ ∈ C ∞ such that ˜ χ ≡ σ ( H +0 ) ∪ σ ( H − k ). First, we compute L X j =0 (1 − χ Re E )( λ j ) | ϕ j (0) | λ j − E − Z R (1 − χ Re E )( λ ) dN +0 ( λ ) λ − E = h δ , [ ˜ χ (1 − χ Re E )] ( H L )( H L − E ) − δ i− h δ , [ ˜ χ (1 − χ Re E )] ( H +0 )( H +0 − E ) − δ i , L X j =0 (1 − χ Re E )( λ j ) | ϕ j ( L ) | λ j − E − Z R (1 − χ Re E )( λ ) dN − k ( λ ) λ − E = h δ L , [ ˜ χ (1 − χ Re E )] ( H L )( H L − E ) − δ L i− h δ L , [ ˜ χ (1 − χ Re E )] ( H − k )( H − k − E ) − δ L i , and L X j =0 (1 − χ Re E )( λ j ) ϕ j ( L ) ϕ j (0) λ j − E = h δ L , [ ˜ χ (1 − χ Re E )] ( H L )( H L − E ) − δ i . By the definition of χ Re E , the function λ ( λ − E ) − ˜ χ ( λ )(1 − χ Re E )( λ ) is C ∞ on R ; moreover,its semi-norms (see (4.14)) are bounded uniformly in E ∈ C . Thus, there exists an almost analyticextension of [ ˜ χ (1 − χ Re E )]( · )( · − E ) − such that, uniformly in E , one has (4.14).In the same way as we obtained (4.16), we obtain(5.6) (cid:12)(cid:12)(cid:12)D δ L , h ( ˜ H L − z ) − − ( H − k − z ) − i δ L E(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:10) δ , (cid:2) ( H L − z ) − − ( H +0 − z ) − (cid:3) δ (cid:11)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:10) δ , ( H L − z ) − δ L (cid:11)(cid:12)(cid:12) ≤ C | Im z | e − L | Im z | /C Plugging (5.6) into (4.15) and using (4.14) for [ ˜ χ (1 − χ Re E )]( · )( · − E ) − , we get ∀ K ∈ N , sup L ≥ L ≡ k mod ( p ) L K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X j =0 (1 − χ Re E )( λ j ) | ϕ j (0) | λ j − E − Z R (1 − χ Re E )( λ ) dN +0 ( λ ) λ − E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < + ∞ This entails (5.4) and completes the proof of Lemma 5.1. (cid:3) Let us now estimate the part of Γ L ( E ) associated to the Dirichlet eigenvalues close to Re E . There-fore, define(5.7) Γ χL ( E ) = L X j =1 χ Re E ( λ j ) λ j − E (cid:18) | ϕ j ( L ) | ϕ j (0) ϕ j ( L ) ϕ j (0) ϕ j ( L ) | ϕ j (0) | (cid:19) . We prove ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 51 Lemma 5.2. There exists ε > such that, for N ≥ , there exists C N such that, for L sufficientlylarge such that L ≡ k mod ( p ) , one has sup Re E ∈ I − ε< Im E< (cid:12)(cid:12)(cid:12) Γ χL ( E ) − Γ eff L ( E ) + ˜ M ( E ) (cid:12)(cid:12)(cid:12) ≤ C N L − N where ˜ M is defined in (5.5) . Clearly Lemmas 5.1 and 5.2 immediately yield Theorem 5.1. Proof of Lemma 5.2. Recall that the quasi-momentum θ p defines a real analytic one-to-one mono-tonic map from the interior of each band of spectrum onto the set (0 , π ), ( − π, 0) or ( − π, π ) (de-pending on the spectral band containing I + [ − ε, ε ] where ε > θ ′ p is positive in the interior of a spectral band. Thus, for L sufficiently large, the real part of the derivative θ ′ p,L (see (4.2)) is positive I + [ − ε, ε ] and θ p,L isreal analytic one-to-one on a complex neighborhood of ( I + [ − ε, ε ]) + i [ − ε, ε ] (possibly at theexpense of reducing ε somewhat).By (2.9), (4.8) and Theorem 4.2, one may write(5.8) Γ χL ( E ) = 1 L − k X j ∈ Z χ Re E (cid:16) θ − p,L (cid:16) πjL − k (cid:17)(cid:17) θ − p,L (cid:16) πjL − k (cid:17) − E M (cid:18) θ − p,L (cid:18) πjL − k (cid:19)(cid:19) where(5.9) M ( λ ) := (cid:18) f k,L ( λ ) σ I e i ( L − k ) θ p,L ( λ ) p f k,L ( λ ) f ,L ( λ ) σ I e i ( L − k ) θ p,L ( λ ) p f k,L ( λ ) f ,L ( λ ) f ,L ( λ ) (cid:19) . and the matrix M is analytic in the rectangle ( I + [ − ε, ε ]) + i [ − ε, ε ]. Thus, the Poisson formulatells us that Γ χL ( E ) = 1 L − k X j ∈ Z Z R e − iπjx χ Re E (cid:16) θ − p,L (cid:16) πxL − k (cid:17)(cid:17) θ − p,L (cid:16) πxL − k (cid:17) − E M (cid:18) θ − p,L (cid:18) πxL − k (cid:19)(cid:19) dx = X j ∈ Z π Z R e − ij ( L − k ) θ p,L ( λ ) χ Re E ( λ ) λ − E θ ′ p,L ( λ ) M ( λ ) dλ = X j ∈ Z π Z R M j,χ ( E, λ, λ ) dλ (5.10)by the definition of χ Re E ; here, we have set M j,χ ( E, λ, β ) := e − ij ( L − k ) θ p,L ( β +Re E ) χ ( λ ) β − i Im E θ ′ p,L ( β + Re E ) M ( β + Re E ) . Let us now study the individual terms in the last sum in (5.10). Therefore, recall that, on [ − ε, ε ], χ is identically 1 and that λ θ p,L ( λ +Re E ) and λ M ( λ ) are analytic in ( I +[ − ε, ε ])+ i [ − ε, ε ];moreover, by (4.3), for some δ > 0, one has(5.11) lim inf L → + ∞ inf λ ∈ [ − ε, ε ] θ ′ p,L ( λ + Re E ) ≥ lim inf L → + ∞ inf E ∈ I θ ′ p,L ( E ) ≥ δ. Recall also that Im E < 0. Consider ˜ χ : R → [0 , 1] smooth such that ˜ χ = 1 on [ − ε, ε ] and ˜ χ = 0outside [ − ε, ε ].In the complex plane, consider the paths γ ± : R → C defined by γ ± ( λ ) = λ ± iε ˜ χ ( λ ) . As − ε ≤ Im E < 0, by contour deformation, we have Z R M j,χ ( E, λ, λ ) dλ = Z R M j,χ ( E, λ, γ + ( λ )) dλ, Z R M j,χ ( E, λ, λ ) dλ = − iπe − ij ( L − k ) θ p,L ( E ) θ ′ p,L ( E ) M ( E ) + Z R M j,χ ( E, λ, γ − ( λ )) dλ. We then estimate • for j < 0, using a non-stationary phase argument as the integrand is the product of a smoothfunction with an rapidly oscillating function (using | j | ( L − k ) as the large parameter), onethen estimates Z R M j,χ ( E, λ, γ + ( λ )) dλ = O (cid:0) ( | j | L ) −∞ (cid:1) . The phase function is complex but its real part is non positive as Im θ p,L ( γ + ( · ) + Re E ) ≥ χ (by (5.11)). Note that the off-diagonal terms of M ( λ ) also carry arapidly oscillating exponential (see (5.9)) but it clearly does not suffice to counter the mainone. • in the same way, for j > 0, one has Z R M j,χ ( E, λ, γ − ( λ )) dλ = O (cid:0) ( | j | L ) −∞ (cid:1) . Thus, we compute for j < Z R M j,χ ( E, λ, λ ) dλ = O (cid:0) ( | j | L ) −∞ (cid:1) , (5.12) for j > Z R M j,χ ( E, λ, λ ) dλ = − iπe − ij ( L − k ) θ p,L ( E ) θ ′ p,L ( E ) M ( E ) + O (cid:0) ( | j | L ) −∞ (cid:1) . (5.13)Finally, for j = 0, the contour deformation along γ + yields Z R χ ( λ ) λ − i Im E M ( λ + Re E ) dλ = Z R χ Re E ( λ ) λ − E θ ′ p,L ( λ ) (cid:18) f k,L ( λ ) 00 f ,L ( λ ) (cid:19) dλ + O (cid:0) L −∞ (cid:1) = Z R χ Re E ( λ ) λ − E (cid:18) dN − k ( λ ) 00 dN +0 ( λ ) (cid:19) + O (cid:0) L −∞ (cid:1) by Corollary 4.1.Plugging this, (5.12) and (5.13) into (5.10) and computing the geometric sum immediately yields thefollowing asymptotic expansion (where the remainder term is uniform on the rectangle I + i [ − ε, χL ( E ) = − i X j> e − ij ( L − k ) θ p,L ( E ) θ ′ p,L ( E ) M ( E )+ Z R χ Re E ( λ ) λ − E (cid:18) dN − k ( λ ) 00 dN +0 ( λ ) (cid:19) + O (cid:0) L −∞ (cid:1) = − e − i ( L − k ) θ p,L ( E ) sin(( L − k ) θ p,L ( E )) θ ′ p,L ( E ) M ( E )+ Z R χ Re E ( λ ) λ − E (cid:18) dN − k ( λ ) 00 dN +0 ( λ ) (cid:19) + O (cid:0) L −∞ (cid:1) . (5.14)This completes the proof of Lemma 5.2. (cid:3) ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 53 The proof of Theorem 5.2. To prove (5.1), for Im E < − ε , it suffices to write L X j =0 | ϕ j (0) | λ j − E − Z R dN +0 ( λ ) λ − E = h δ , ( H L − E ) − δ i − h δ , ( H +0 − E ) − δ i = h δ , ( H L − E ) − δ L ih δ L +1 , ( H +0 − E ) − δ i and L X j =0 | ϕ j ( L ) | λ j − E − Z R dN − k ( λ ) λ − E = h δ , ( H L − E ) − δ L ih δ L +1 , ( H − k − E ) − δ i , L X j =0 ϕ j ( L ) ϕ j (0) λ j − E = h δ L , ( H L − E ) − δ i and to use the Combes-Thomas estimate (5.6). This completes the proof of Theorem 5.2. (cid:3) The proofs of Theorems 1.2, 1.3 and 1.4. We will now use Theorems 5.1 and 5.2 to proveTheorems 1.2, 1.3 and 1.4.5.2.1. The proof of Theorem 1.2. The first statement of Theorem 1.2 is an immediate consequenceof the characteristic equations for the resonances (2.4) and (2.8) and the description of the eigen-values of H L given in Theorem 4.2.When • = N , i.e., for the operator on the half-line, if I ⊂ ( − , 2) does not meet Σ N , there exists C > L sufficiently large dist( I, σ ( H L )) > /C . Thus, on the set I − i [0 , + ∞ ), one hasIm S L ( E ) ≤ Im E/C . As on I , one has Im θ p ( E ) > /C (see section 2), the characteristic equa-tion (2.4) admits a solution E such that Re E ∈ I only if Im E < /C . This completes the proofof point (1) of Theorem 1.2 for • = N .For • = Z , i.e., to study equation (2.8), one reasons in the same way except that one replacesthe study of S L ( E ) by that of h Γ L ( E ) u, u i for u an arbitrary vector in C of unit length. Thiscompletes the proof of point (1) of Theorem 1.2Point (3a) is an immediate consequence of Theorems 3.3 and 3.2 and the description of the eigenval-ues of H L outside Σ Z . Notice that in the present case d j in Theorems 3.3 and 3.2 is bounded frombelow by a constant independent of L and a • j is exponentially small and described by Theorem 4.2.Point (3b) is an immediate consequence of the description of the eigenvalues of H L outside Σ Z incase (3) of Theorem 5.2 and Theorem 3.1. Indeed, in the present case d j and and a • j are both oforder 1; thus, Theorem 3.1 guarantees, around the common eigenvalue for H − k and H +0 , a rectangleof width of order 1 free of resonances.Let us now turn to the proof of point (2). Therefore, we first prove the following corollary ofTheorem 5.1 Corollary 5.1. Fix I ⊂ ◦ Σ Z compact. There exists η > such that, for L sufficiently large, onehas (5.15) min Re E ∈ I Im E ∈ [ − η /L, (cid:12)(cid:12)(cid:12) S L ( E ) + e − iθ ( E ) (cid:12)(cid:12)(cid:12) ≥ η and min Re E ∈ I Im E ∈ [ − η /L, (cid:12)(cid:12)(cid:12) det (cid:16) Γ L ( E ) + e − iθ ( E ) (cid:17)(cid:12)(cid:12)(cid:12) ≥ η . Clearly, Corollary 5.1 implies that neither equation (2.4) nor equation (2.8) can have a solution in I + i ] − η /L, (cid:3) Before proving Corollary 5.1, we first prove Propositions 5.2 and 5.3 as these will be used in theproof of Corollary 5.1. Results on the auxiliary functions defined in section 1.2.2. Recall that N − k is defined insection 1.2.2. We prove Proposition 5.1. For k ∈ { , · · · , p − } , dN − k is a positive measure that is absolutely continuouson Σ Z . Moreover, its density, say, E n − k ( E ) is real analytic on ◦ Σ Z and there exists f − k : ◦ Σ Z → R a positive real analytic function such that, on ◦ Σ Z , one has n − k ( E ) = f − k ( E ) n ( E ) .Proof. Proposition 5.1 is an immediate consequence of Theorems 5.1 and 5.2 and Corollary 4.1. (cid:3) For Ξ − k defoined in (1.5), we prove Proposition 5.2. Ξ − k vanishes identically if and only if V ≡ , i.e., V vanishes identically. More-over, if V then there exists ξ − k = 0 and α − k ∈ { , , · · · } such that Ξ − k ( E ) ∼ | E |→∞ Im E< ξ − k E − α − k .Proof. We will do the proofs for the function Ξ − k . Proposition 5.2 is an immediate consequence ofthe fact that, in the lower half-plane, the function E 7→ − e − i arccos( E/ = − E − r E − N and the vector δ ; this follows from a directcomputation (see Remark 2.1 and (2.2) for n = 0). Now, if one lets W be the symmetric of τ k V withrespect to 0, the spectral measure dN − k is also the spectral measure of the Schr¨odinger operator H k = − ∆ + W on N associated to δ . The equality of the Borel transforms implies the equality ofthe measures but δ is cyclic for both operators so the operators have equal spectral measures. Thisimplies that the two operators are equal and, thus, the symmetric of τ k V has to vanish identicallyon N . As V is periodic, V must vanish identically.As for the second point, if the function Ξ − k were to vanish to infinite order at E = − i ∞ , as each ofthe terms Z R dN − k ( λ ) λ − E and − E − r E − E − , these two expansions would be equal. The n -th coefficient of these expansion are respectivelythe n -th moments of the spectral measures of H k and − ∆ +0 (associated to the cyclic vector δ ). Sothese moments would coincide and, thus, the spectral measures would coincide. One concludes asabove. (cid:3) (cid:3) For c • defined in (1.6) and (1.7), we prove Proposition 5.3. Pick • ∈ { N , Z } . Let I ⊂ ( − , ∩ ◦ Σ Z be a compact interval.There exists a neighborhood of I such that, in this neighborhood, the function E c • ( E ) is analyticand has a positive imaginary part.The function c N (resp. c Z ) takes the value i only at the zeros of Ξ − k (resp. Ξ − k Ξ +0 ).Proof. On { Im E < } , define the functions g − k ( E ) := i + Ξ − k ( E ) π n − k ( E ) = 1 π n − k ( E ) (cid:16) S − k ( E ) + e − i arccos( E/ (cid:17) , (5.16) g +0 ( E ) := i + Ξ +0 ( E ) π n +0 ( E ) = 1 π n +0 ( E ) (cid:16) S +0 ( E ) + e − i arccos( E/ (cid:17) . (5.17)First, the analyticity of g − k and g +0 is clear; indeed, all the functions involved are analytic and thefunctions n +0 and n − k stay positive on ◦ Σ Z . Moreover, these functions can be analytically continued ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 55 through ( − , ∩ ◦ Σ Z . By (1.4), for E real, one has Im g − k ( E ) =Im g +0 ( E ) = Im e − iθ ( E ) which ispositive (see section 2). Thus, the functions E g − k ( E ) and E g +0 ( E ) do not vanish on I .Moreover, as(5.18) g +0 ( E ) g − k ( E ) − g +0 ( E ) + g − k ( E ) = − g +0 ( E ) + g − k ( E ) + 11 g +0 ( E ) + 1 g − k ( E ) ;this function has a positive imaginary part on I .This proves the first two properties of c • stated in Proposition 5.3. By the very definition of c • and g − k , the last property stated in Proposition 5.3 is obviously satisfied in the case of the half-line; forthe full line , i.e., if • = Z , the last property is a consequence of the following computation c Z ( E ) − i = g +0 ( E ) g − k ( E ) − g +0 ( E ) + g − k ( E ) − i = ( g +0 ( E ) − i )( g − k ( E ) − i ) g +0 ( E ) + g − k ( E )= Ξ +0 ( E )Ξ − k ( E )2 iπ n +0 ( E ) n − k ( E ) + πn − k ( E )Ξ +0 ( E ) + πn +0 ( E )Ξ − k ( E ) . (5.19)This completes the proof of Proposition 5.3. (cid:3) (cid:3) The proof of Corollary 5.1. In view of Theorem 5.1, to obtain (5.15), it suffices to prove thatthere exists η > L sufficiently large, one hasmin Re E ∈ I Im E ∈ [ − η /L, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ ′ p,L ( E ) f − k ( E ) e − iu L ( E ) sin u L ( E ) − Z R dN − k ( λ ) λ − E − e − iθ ( E ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ η where u L ( E ) := ( L − k ) θ p,L ( E ).We compute(5.20) θ ′ p,L ( E ) f − k ( E ) e − iu L ( E ) sin u L ( E ) − Z R dN − k ( λ ) λ − E − e − iθ ( E ) = θ ′ p,L ( E ) f − k ( E ) (cid:0) cot u L ( E ) − g − k ( E ) (cid:1) where g − k is defined in (5.16). Thus, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ ′ p,L ( E ) f − k ( E ) e − iu L ( E ) sin u L ( E ) − Z R dN − k ( λ ) λ − E − e − iθ ( E ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & (cid:12)(cid:12) cot u L ( E ) − g − k ( E ) (cid:12)(cid:12) as, for η sufficiently small and L ≥ 1, one has0 < min Re E ∈ I Im E ∈ [ − η/L, (cid:12)(cid:12) θ ′ p,L ( E ) f − k ( E ) (cid:12)(cid:12) ≤ max Re E ∈ I Im E ∈ [ − η/L, (cid:12)(cid:12) θ ′ p,L ( E ) f − k ( E ) (cid:12)(cid:12) < + ∞ . Now, notice that, by Corollary 4.1, for E ∈ I , one has(5.21) Im (cid:18)Z R dN − k ( λ ) λ − E (cid:19) = − θ ′ p,L ( E ) f − k ( E ) = − π n − k ( E ) . Thus, as E Im e − iθ ( E ) is positive on I , the analytic function E g − k ( E ) has positive imaginarypart larger than, say, 2˜ η on I ; hence, it has imaginary part larger than, say, ˜ η in some neighborhoodof I + D (0 , η ) (for sufficiently small η > M be the maximum modulus of this function on I + D (0 , η ). Thus, as max Re E ∈ I Im E ∈ [ − η /L, | θ ′ p,L ( E ) | . 1, one hasmax Re E ∈ I Im E ∈ [ − η /L, | cot( u L ( E )) | < M | Im cot u L ( E ) | . ( M + 1) η . Possibly reducing η , this guarantees that, for Re E ∈ I and Im E ∈ [ − η /L, (cid:12)(cid:12) cot u L ( E ) − g − k ( E ) (cid:12)(cid:12) ≥ M − M ≥ M or Im (cid:0) cot u L ( E ) − g − k ( E ) (cid:1) ≤ − ˜ η + ˜ η/ − ˜ η/ . This completes the proof of the first lower bound in (5.15) in Corollary 5.1.To prove the second bound in (5.15), using (5.2), we computedet (cid:0) Γ eff L ( E ) + e − iθ ( E ) (cid:1) n − k ( E ) n +0 ( E ) = (cid:0) cot u L ( E ) − g − k ( E ) (cid:1) (cid:0) cot u L ( E ) − g +0 ( E ) (cid:1) − u L ( E )= − (cid:0) g +0 ( E ) + g − k ( E ) (cid:1) (cid:18) cot u L ( E ) − g +0 ( E ) g − k ( E ) − g +0 ( E ) + g − k ( E ) (cid:19) (5.22)where g − k and g +0 are defined by (5.16) and (5.17).Using Proposition 5.3, one then concludes the non-vanishing of E det (cid:0) Γ eff L ( E ) + e − iθ ( E ) (cid:1) in thecomplex rectangle { Re E ∈ I, Im E ∈ [ − η /L, } (for η sufficiently small) in the same way asabove. This completes the proof of Corollary 5.1. (cid:3) The proof of Theorem 1.3. To solve (2.4) and (2.8), by Theorem 5.1, we respectively firstsolve the equations(5.23) θ ′ p,L ( E ) f − k ( E ) e − iu L ( E ) sin u L ( E ) = Z R dN − k ( λ ) λ − E − e − iθ ( E ) and det (cid:16) Γ eff L ( E ) + e − iθ ( E ) (cid:17) = 0in a rectangle I + i [ − η, − ˜ η/L ]. Indeed, in such a rectangle, by Theorem 5.1, equations (2.4) and (2.8)are respectively equivalent to(5.24) θ ′ p,L ( E ) f − k ( E ) e − iu L ( E ) sin u L ( E ) = Z R dN − k ( λ ) λ − E − e − iθ ( E ) + O (cid:0) L −∞ (cid:1) and det (cid:16) Γ eff L ( E ) + e − iθ ( E ) (cid:17) = O (cid:0) L −∞ (cid:1) where the terms O ( L −∞ ) are analytic in a rectangle ˜ I + i [ − η, − 0) (where I ⊂ ˜ I ) and the bound O ( L −∞ ) holds in the supremum norm.Thanks to (5.20) for • = N and to (5.22) for • = Z , to solve the equations (5.23), it suffices to solve(5.25) cot u L ( E ) = c • ( E )where we recall u L ( E ) := ( L − k ) θ p,L ( E ) and, g +0 and g − k being respectively defined in (5.17)and (5.16), and, as in section 1.2.3, one has set • c N ( E ) := g − k ( E ) in the case of the half-line, • c Z ( E ) := g +0 ( E ) g − k ( E ) − g +0 ( E ) + g − k ( E ) in the case of the line.We want to solve (5.25) is a rectangle I + i [ − ε, 0) for some ε small but fixed. Using Proposition 5.3,we pick ε so small that, in the rectangle I + i [ − ε, c • − i are those on the real ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 57 line and Im c • is positive in I + i [ − ε, u = ( L − k ) θ p,L ( E ) that is, we write E = θ − p,L (cid:18) uL − k (cid:19) . As, for L sufficiently large, inf L ≥ L E ∈ I + i [ − ε, Re θ ′ p,L ( E ) > c > 0, at the cost of possibly reducing ε , thisreal analytic change of variables maps I + [ − ε, ε ] + i [ − ε, 0) into, say, D L such that I L + i [ − η ( L − k ) , ⊂ D L (for some η > 0) where I L = ( L − k ) θ p,L ( I + [ − ε/ , ε/ I L + i [ − η ( L − k ) , 0] into some domain, say, ˜ D L such that I + [ − ε ′ , ε ′ ] + i [ − ε ′ , ⊂ ˜ D L (forsome 0 < ε ′ < ε ). Now, to find all the solutions to (5.25) in I + i [ − ε ′ , I L + i [ − η ( L − k ) , u = c • ◦ θ − p,L (cid:18) uL − k (cid:19) As u cot u is π periodic, we split I L + i [ − η ( L − k ) , 0] into vertical strips of the type lπ +[0 , π ] + i [ − η ( L − k ) , l − ≤ l ≤ l + , ( l − , l + ) ∈ Z . Without loss of generality, we may assume that I L = [ l − , l + ] π . To solve (5.26) on the rectangle lπ + [0 , π ] + i [ − η ( L − k ) , u by lπ andsolve the following equation on [0 , π ] + i [ − η ( L − k ) , u = c • l,L ( u ) where c • l,L ( · ) := c • ◦ θ − p,L (cid:18) · + lπL − k (cid:19) . In proving Theorem 1.2, we have already shown that for some ˜ η > L sufficientlylarge and l − ≤ l ≤ l + ), (5.27) does not have a solution in [0 , π ] + i [ − ˜ η, , π ) + i ( −∞ , 0] to C + \ { i } . Thus, for L sufficiently large and˜ η sufficiently small, the cotangent defines a one-to-one mapping from [0 , π ) + i [ − η ( L − k ) , − ˜ η ] onto T L = D ( z + , r + ) \ D ( z − , r − ), analytic in the interior of [0 , π ) + i [ − η ( L − k ) , − ˜ η ] and continuous upto the boundary where we have defined z + = i e η ( L − k ) + 1 e η ( L − k ) − , z − = i e η − e η − , r + = 2 e η e η − , r − = 2 e η ( L − k ) e η ( L − k ) − . Moreover, the boundaries { } + i [ − η ( L − k ) , − ˜ η ] and { π } + i [ − η ( L − k ) , − ˜ η ] are mapped onto theinterval [ z − + ir − , z + + ir + ].Let ˜ Z • denote the finite set of zeros of E c • ( E ) − i in I . Then, by a Taylor expansion near thezeros of c − i , we know that, for η sufficiently small, there exists ε > k ≥ L sufficiently large, • for ε ∈ (0 , ε ), there exists 0 < η − such that, for l − ≤ l ≤ l + , if ∀ ˜ E ∈ ˜ Z • , one has (cid:12)(cid:12)(cid:12)(cid:12) θ − p,L (cid:18) lπL − k (cid:19) − ˜ E (cid:12)(cid:12)(cid:12)(cid:12) ≥ ε then ∀ u ∈ [0 , π ] + i [ − η ( L − k ) , η − ≤ | Im c • l,L ( u ) − | ; • for u ∈ [0 , π ] + i [ − η ( L − k ) , 0] and ˜ E the point in ˜ Z • closest to θ − p,L (cid:18) lπL − k (cid:19) , one has(5.28) ε ≤ (cid:0) − Im c • l,L ( u ) (cid:1) · (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) θ − p,L (cid:18) Re u + lπL − k (cid:19) − ˜ E (cid:12)(cid:12)(cid:12)(cid:12) + | Im u | L − k (cid:21) − ˜ k ≤ ε where ˜ k is the order of ˜ E as a zero of E c • ( E ) − i .As a consequence of the above description of c • l,L , we obtain Lemma 5.3. There exists ˜ η and η small such that, for L sufficiently large, for all l − ≤ l ≤ l + , u c • l,L ( u ) maps the rectangle [0 , π ]+ i [ − η ( L − k ) , − ˜ η ] into a compact subset of D ( z + , r + ) \ D ( z − , r − ) in such a way that (5.29) sup u ∈ ∂ ([0 ,π ]+ i [ − η ( L − k ) , − ˜ η ]) (cid:12)(cid:12) cot u − c • l,L ( u ) (cid:12)(cid:12) & (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ˜ E − θ − p,L (cid:18) lπL − k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + ˜ ηL − k (cid:19) ˜ k where ˜ E is the root of E c • ( E ) − i closest to θ − p,L (cid:18) lπL − k (cid:19) and ˜ k is the order of this root. Note that, under the assumptions of Lemma 5.3, (5.29) implies thatsup u ∈ ∂ ([0 ,π ]+ i [ − η ( L − k ) , − ˜ η ]) (cid:12)(cid:12) cot u − c • l,L ( u ) (cid:12)(cid:12) & L − ˜ k Thus, we can define the analytic mapping cot − ◦ c • l,L on [0 , π ] + i [ − η ( L − k ) , − ˜ η ]; it maps therectangle [0 , π ] + i [ − η ( L − k ) , − ˜ η ] into a compact subset of (0 , π ) + i ( − η ( L − k ) , − ˜ η ). The equa-tion (5.27) on [0 , π ] + i [ − η ( L − k ) , − ˜ η ] is, thus, equivalent to the following fixed point equation onthe same rectangle(5.30) u = cot − ◦ c • l,L ( u )We note that, for α ∈ (0 , L sufficiently large, if for some ˜ E ∈ ˜ Z • of multiplicity ˜ k , one has (cid:12)(cid:12)(cid:12)(cid:12) θ − p,L (cid:18) lπL − k (cid:19) − ˜ E (cid:12)(cid:12)(cid:12)(cid:12) < L − α then, equation (5.27) has no solution in [0 , π ] + i [ − η ( L − k ) , − ˜ η ] outsideof the set R l,L := [0 , π ] + i " − η ( L − k ) , α ˜ k (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) θ − p,L (cid:18) lπL − k (cid:19) − ˜ E (cid:12)(cid:12)(cid:12)(cid:12) + 1 L (cid:21) . Indeed, for u ∈ ([0 , π ] + i [ − η ( L − k ) , − ˜ η ]) \ R l,L , by (5.28), that is, for0 ≤ Re u ≤ π and − α ˜ k L ≤ α ˜ k (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) θ − p,L (cid:18) lπL − k (cid:19) − ˜ E (cid:12)(cid:12)(cid:12)(cid:12) + 1 L (cid:21) ≤ Im u ≤ − ˜ η one has (cid:12)(cid:12)(cid:12) c • l,L ( u ) − i (cid:12)(cid:12)(cid:12) . L − α ˜ k and | cot u − i | & L − α ˜ k/ .So, if for some ˜ E ∈ ˜ Z • , one has (cid:12)(cid:12)(cid:12)(cid:12) θ − p,L (cid:18) lπL − k (cid:19) − ˜ E (cid:12)(cid:12)(cid:12)(cid:12) < L − α , it suffices to solve (5.30) on R l,L . Wecompute the derivative of c • l,L in the interior of R l,L ddu (cid:0) cot − ◦ c • l,L (cid:1) ( u ) = − L − k c ′ ◦ θ − p,L (cid:16) u + lπL − k (cid:17) (cid:16) c • l,L ( u ) (cid:17) · θ ′ p,L (cid:16) θ − p,L (cid:16) u + lπL − k (cid:17)(cid:17) = 1 L − k c ′ ◦ θ − p,L (cid:16) u + lπL − k (cid:17) c • l,L ( u ) − i · c • l,L ( u ) + i · θ ′ p,L (cid:16) θ − p,L (cid:16) u + lπL − k (cid:17)(cid:17) . Thus, fixing α ∈ (0 , ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 59 • if l is such that, for some ˜ E ∈ ˜ Z • , one has (cid:12)(cid:12)(cid:12)(cid:12) θ − p,L (cid:18) lπL − k (cid:19) − ˜ E (cid:12)(cid:12)(cid:12)(cid:12) < L − α , for u ∈ R l,L , weestimate (cid:12)(cid:12)(cid:12)(cid:12) ddu (cid:0) cot − ◦ c • l,L (cid:1) ( u ) (cid:12)(cid:12)(cid:12)(cid:12) . L − k (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) θ − p,L (cid:18) lπL − k (cid:19) − ˜ E (cid:12)(cid:12)(cid:12)(cid:12) + | Im u | L − k (cid:21) − . L − k ) (cid:12)(cid:12)(cid:12) θ − p,L (cid:16) lπL − k (cid:17) − ˜ E (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) log h(cid:12)(cid:12)(cid:12) θ − p,L (cid:16) lπL − k (cid:17) − ˜ E (cid:12)(cid:12)(cid:12) + ˜ ηL − k i(cid:12)(cid:12)(cid:12) . L ;(5.31) • if l is such that, for all ˜ E ∈ ˜ Z • , one has (cid:12)(cid:12)(cid:12)(cid:12) θ − p,L (cid:18) lπL − k (cid:19) − ˜ E (cid:12)(cid:12)(cid:12)(cid:12) ≥ L − α , for u ∈ [0 , π ]+ i [ − η ( L − k ) , − ˜ η ], we estimate (cid:12)(cid:12)(cid:12)(cid:12) ddu (cid:0) cot − ◦ c • l,L (cid:1) ( u ) (cid:12)(cid:12)(cid:12)(cid:12) . L − k (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) θ − p,L (cid:18) lπL − k (cid:19) − ˜ E (cid:12)(cid:12)(cid:12)(cid:12) + | Im u | L − k (cid:21) − . L − k ) (cid:12)(cid:12)(cid:12) θ − p,L (cid:16) lπL − k (cid:17) − ˜ E (cid:12)(cid:12)(cid:12) . L − α . (5.32)Hence, for L sufficiently large, cot − ◦ c • l,L is a contraction on R l,L . Equation (5.30) thus admits aunique solution, say, ˜ u • l,L in the rectangle [0 , π ] + i [ − η ( L − k ) , − ˜ η ]. This solution is a simple root of u u − cot − ◦ c • l,L ( u ). Hence, ˜ u • l,L is the only solution to equation (5.27) in [0 , π ]+ i [ − η ( L − k ) , − ˜ η ].By (5.24), for L sufficiently large, for l − ≤ l ≤ l + , both the equations(5.33) S L ◦ θ − p,L (cid:18) u + lπL − k (cid:19) + e − iθ ( θ − p,L ( u + lπL − k ) ) = 0 anddet (cid:18) Γ L ◦ θ − p,L (cid:18) u + lπL − k (cid:19) + e − iθ ( θ − p,L ( u + lπL − k ) ) (cid:19) = 0can be rewritten as(5.34) u = cot − (cid:0) c • l,L ( u ) + O (cid:0) L −∞ (cid:1)(cid:1) = cot − ◦ c • l,L ( u ) + O (cid:0) L −∞ (cid:1) in [0 , π ] + i [ − η ( L − k ) , − ˜ η ].Thus, each of the equations in (5.33) admits a single solution in [0 , π ] + i [ − η ( L − k ) , − ˜ η ] andthis root is simple; moreover, this solution, say, u l,L satisfies (cid:12)(cid:12)(cid:12) u • l,L − ˜ u • l,L (cid:12)(cid:12)(cid:12) = O ( L −∞ ); indeed, thebounds (5.31) and (5.32) guarantee that one can apply Rouch´e’s Theorem on the disk D (˜ u • l,L , L − k )for any k ≥ Lemma 5.4. Pick I as above. Then, there exists η > such that, for L sufficiently large s.t. L = N p + k , the resonances in I + i [ − η, are the energies ( z • l ) l − ≤ l ≤ l + defined by (5.35) z • l = θ − p,L (cid:18) u • l,L + lπL − k (cid:19) belonging to I + i [ − η, . Let us complete the proof of Theorem 1.7 that is, prove that, for η sufficiently small, for L sufficientlylarge such that L ≡ k mod ( p ), is the unique resonance in (cid:20) Re (˜ z • l + ˜ z • l − )2 , Re (˜ z • l + ˜ z • l +1 )2 (cid:21) + i [ − η, z • l is defined in (1.9).Therefore, we first note that the Taylor expansion of θ − p,L , (4.1) and the quantization condition (4.3)imply that z • l = λ l + 1 πn ( λ l ) L u • l,L + O (cid:18) log LL (cid:19) ! as Re u l,L ∈ [0 , π ) and − log L . Im u l,L . − c • l,L ( u ) = c • (cid:20) λ l + uπ n ( λ l ) L + O (cid:18) u L (cid:19)(cid:21) using (1.9) and (5.35), we compute z • l − ˜ z • l = 1 πn ( λ l ) L (cid:18) u • l,L − cot − ◦ c • (cid:20) λ l + 1 π n ( λ l ) L cot − ◦ c • (cid:18) λ l − i log LL (cid:19)(cid:21)(cid:19) + O (cid:18) log LL (cid:19) ! . Thus, one has z • l − ˜ z • l = 1 πn ( λ l ) L (cid:0) u • l,L − cot − ◦ c • l,L (cid:2) cot − ◦ c • l,L ( − iπ n ( λ l ) log L ) (cid:3)(cid:1) + O (cid:18) log LL (cid:19) ! . As u l,L solves (5.34), using (5.31) and (5.32), we thus obtain that | z • l − ˜ z • l | . L log L (cid:12)(cid:12) u • l,L − cot − ◦ c • l,L ( − iπ n ( λ l ) log L ) (cid:12)(cid:12) + (cid:18) log LL (cid:19) . (cid:12)(cid:12)(cid:12) u • l,L (cid:12)(cid:12)(cid:12) + log LL log L + (cid:18) log LL (cid:19) . L log L using again Re u l,L ∈ [0 , π ) and − log L . Im u l,L . − (cid:3) The proofs of Propositions 1.1 and 1.2. Proposition 1.2 is an immediate consequence ofTheorem 1.3, the definition of ˜ z • l (1.9) and the standard asymptotics of cot near − i ∞ , i.e., cot z = i + 2 ie − iz + O (cid:0) e − iz (cid:1) .To prove Proposition 1.1, it suffices to notice that, under the assumptions of Proposition 1.1, thebound (5.32) on the derivative of cot − ◦ c • l,L on the the rectangle R l,L becomes (cid:12)(cid:12)(cid:12)(cid:12) ddu (cid:0) cot − ◦ c • l,L (cid:1) ( u ) (cid:12)(cid:12)(cid:12)(cid:12) . L . Thus, as a solution to (5.30), u • l,L admits an asymptotic expansion in inverse powers of L . Pluggingthis into (5.35) yields the asymptotic expansion for the resonance. Then, (1.11) follows from thecomputation of the first terms. (cid:3) The proof of Theorem 1.4. Theorem 1.4 is an immediate consequence of Theorem 5.2, thefact that the functions are analytic in the lower complex half-plane and have only finitely manyzeros there and the argument principle. (cid:3) The half-line periodic perturbation: the proof of Theorem 1.5. Using the same no-tations as above, we can write H ∞ = (cid:18) H −− | δ − ih δ || δ ih δ − | − ∆ +0 (cid:19) . ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 61 where − ∆ +0 is the Dirichlet Laplacian on ℓ ( N ).Define the operators Γ( E ) := H −− − E − h δ | ( − ∆ +0 − E ) − | δ i | δ − ih δ − | and ˜Γ( E ) := − ∆ +0 − E − h δ − | ( H −− − E ) − | δ − i | δ ih δ | . For Im E = 0, h δ − | ( H −− − E ) − | δ − i and h δ | ( − ∆ +0 − E ) − | δ i have a non vanishing imaginarypart of the same sign; hence, the complex number( h δ | ( − ∆ +0 − E ) − | δ i ) − − h δ − | ( H −− − E ) − | δ − i does not vanish. Thus, by rank one perturbation theory, (see, e.g., [37]), we know that Γ( E ) and˜Γ( E ) are invertible and their inverses are given by(5.36) Γ − ( E ) := ( H −− − E ) − + | H −− − E ) − | δ − ih δ − | ( H −− − E ) − | ( h δ | ( − ∆ +0 − E ) − | δ i ) − − h δ − | ( H −− − E ) − | δ − i . and(5.37) ˜Γ − ( E ) := ( − ∆ +0 − E ) − + | − ∆ +0 − E ) − | δ ih δ | ( − ∆ +0 − E ) − | ( h δ − | ( H −− − E ) − | δ − i − − h δ | ( − ∆ +0 − E ) − | δ i ) . Thus, for Im E = 0, using Schur’s complement formula, we compute(5.38) ( H ∞ − E ) − = (cid:18) Γ( E ) − γ ( E ) γ ∗ (cid:0) E (cid:1) ˜Γ( E ) − (cid:19) . where γ ∗ (cid:0) E (cid:1) is the adjoint of γ (cid:0) E (cid:1) and γ ( E ) := −| Γ( E ) − | δ − ih δ | ( − ∆ +0 − E ) − | . Now, when coming from Im E > − , ∩ ◦ Σ Z , the complex numbers h δ − | ( H −− − E ) − | δ − i and h δ | ( − ∆ +0 − E ) − | δ i keep imaginary parts of the same positive sign;thus, the two operator-valued functions E Γ − ( E ) and E ( H ∞ − E ) − can be analyticallycontinued through ( − , ∩ ◦ Σ Z from the upper to the lower complex half-plane (as operators re-spectively from ℓ ( N ) to ℓ ( N ) and from ℓ ( Z ) to ℓ ( Z )).When coming from the upper half-plane and passing through ( − , \ Σ Z and ◦ Σ Z \ [ − , H ∞ − E ) − . Definition (5.36) and formula (5.38) immediatelyshow that the poles of these continuations only occur at the zeros of the function E − h δ − | ( H −− − E ) − | δ − ih δ | ( − ∆ +0 − E ) − | δ i = 1 − e iθ ( E ) Z R dN − p − ( λ ) λ − E when continued from the upper half-plane through the sets ( − , \ Σ Z and ◦ Σ Z \ [ − , 2] (these setsare finite unions of open intervals).This completes the proof of Theorem 1.5. (cid:3) Resonances in the random case As for the periodic potential, for the random potential, we start with a description of the function E Γ L ( E ) (see (2.9)), that is, with a description of the spectral data for the Dirichlet operator H ω,L . The matrix Γ L in the random case. We recall a number of results on the Dirichlet eigen-values of H ω,L that will be used in our analysis.It is well known that, under our assumptions, in dimension one, the whole spectrum of H ω is inthe localization region (see, e.g., [28, 10, 7]) that is Theorem 6.1. There exists ρ > and α ∈ (0 , such that, one has (6.1) sup L ∈ N ∪{ + ∞} y ∈ J ,L K Im E =0 E X x ∈ J ,L K e ρ | x − y | |h δ x , ( H ω,L − E ) − δ y i| α < ∞ and (6.2) sup L ∈ N ∪{ + ∞} y ∈ J ,L K E X x ∈ J ,L K e ρ | x − y | sup supp f ⊂ R | f |≤ |h δ x , f ( H ω,L ) δ y i| < ∞ . where H ω, + ∞ := H N ω and J , + ∞ K = N . The supremum is taken over the functions f that areBorelian and compactly supported. As a consequence, one can define localization centers e.g. by means of the following results Lemma 6.1 ([13]) . Fix ( l L ) L a sequence of scales, i.e., l L → + ∞ as L → + ∞ . There exists ρ > such that, for L sufficiently large, with probability larger than − e − ℓ L , if (1) ϕ j,ω is a normalized eigenvector of H ω,L associated to E j,ω in Σ , (2) x j ( ω ) ∈ J , L K is a maximum of x 7→ | ϕ j,ω ( x ) | in J , L K ,then, for x ∈ J , L K , one has (6.3) | ϕ j,ω ( x ) | ≤ √ Le ℓ L e − ρ | x − x j ( ω ) | . Note that Lemma 6.1 is of interest only if ℓ L . L ; otherwise (6.3) is obvious. This result can e.g.be applied for the scales l L = 2 log L . In this case, the probability estimate of the bad sets (i.e.when the conclusions of Lemma 6.2 does not hold) is summable. The point x j ( ω ) is a localizationcenter for E j,ω or ϕ j,ω . It is not defined uniquely, but, one easily shows that there exists C > x and x ′ , one has | x − x ′ | ≤ C log L (see [13]). To fixideas, we set the localization center associated to the eigenvalue E j,ω to be the left most maximumof x 7→ k ϕ j,ω k x .We show Lemma 6.2. For any p > , there exists C > and L > (depending on α and p ) such that, for L ≥ L , for any sequence satisfying (1.22) , with probability at least − L − p , there exists at most Cℓ L eigenvalues having a localization center in J , ℓ L K ∪ J L − ℓ L , L K . We will now use the fact that we are dealing with one-dimensional systems to improve upon theestimate (6.3). We prove Theorem 6.2. For any δ > and p ≥ , there exists C > and L > (depending on p and δ )such that, for L ≥ L , with probability at least − L − p , if E j,ω is an eigenvalue in Σ associated tothe eigenfunction ϕ j,ω and the localization center x j,ω then, • if x j,ω ∈ J , L − C log L K , one has (6.4) − ρ ( E j,ω ) − δ ≤ log | ϕ j,ω ( L ) | L − x j,ω ≤ − ρ ( E j,ω ) + δ. ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 63 • if x j,ω ∈ J C log L, L K , one has (6.5) − ρ ( E j,ω ) − δ ≤ log | ϕ j,ω (0) | x j,ω ≤ − ρ ( E j,ω ) + δ. To analyze the resonances of H N ω,L (resp. H Z ω,L ), we shall use (6.4) (resp. (6.4) and (6.5)).We now use these estimates as the starting point of a short digression from the main theme of thispaper. Let us first state a corollary to Theorem 6.2, we prove Theorem 6.3. For any δ > and p ≥ , for L sufficiently large (depending on p and δ ), withprobability at least − L − p , if E j,ω is an eigenvalue in Σ associated to the eigenfunction ϕ j,ω andthe localization center x j,ω then, for | x − x j,ω | ≥ δL and ≤ x ≤ L , one has (6.6) − ρ ( E j,ω ) − δ ≤ log( | ϕ j,ω ( x ) | + | ϕ j,ω ( x − | ) | x − x j,ω | ≤ − ρ ( E j,ω ) + δ. Compare (6.6) to (6.3). There are two improvements. First, the unknown rate of decay ρ is replacedby the Lyapunov exponent ρ ( E j,ω ) which was expected to be the correct decay rate. Indeed, for theone-dimensional discrete Anderson model on the half-axis, it is well known (see, e.g., [5, 7, 33]) that, ω -almost surely, the spectrum is localized and the eigenfunctions decay exponentially at infinity ata rate given by the Lyapunov exponent. In Theorem 6.3, we state that, with a good probability,this is true for finite volume restrictions.Second, in (6.6), we get both an upper and lower bound on the eigenfunction. This is more precisethan (6.3).To our knowledge, such a result was not known until the present paper. The strategy that we useto prove this result can be applied in a more general one-dimensional setting to obtain analoguesof (6.6) (see [24]).We complement this with the much simpler Lemma 6.3. For any C > and p ≥ , there exists K > and L > (depending on I , p and δ )such that, for L ≥ L , with probability at least − L − p , if E j,ω is an eigenvalue in Σ associated tothe eigenfunction ϕ j,ω and the localization center x j,ω then, • if x j,ω ∈ J L − C log L, L K , one has L − K ≤ | ϕ j,ω ( L ) | ; • if x j,ω ∈ J , C log L K , one has L − K ≤ | ϕ j,ω (0) | . The proof of this result is obvious and only uses the fact that the matrices in the cocycle defining theoperator (see section 6.3) are bounded that is, equivalently, that the solutions to the Schr¨odingerequation grow at most exponentially at a rate controlled by the potential.Let us return to the resonances in the random case and the description of the function S L . Recallthat in (2.4), the values ( λ j ) j are the eigenvalues ( E j,ω ) ≤ j ≤ L of H ω,L and the coefficients ( a • j ) j aredefined in Theorem 2.1 and by (2.13). Thus, Theorem 6.2 describes the coefficients ( a • j ) j cominginto S L and Γ L (see (2.4) and (2.8)). Let us now state a few consequences of Theorem 6.2.Fix I a compact interval in Σ the almost sure spectrum of H ω . For • ∈ { N , Z } , define(6.7) d • j,ω = ( L − x j,ω for • = N , min( x j,ω , L − x j,ω ) for • = Z . Taking p > ω almost surely, for δ > L sufficiently large, if λ j = E j,ω ∈ I and d • j,ω ≥ C log L then − ρ ( λ j ) − δ ≤ log a • j d • j,ω ≤ − ρ ( λ j ) + δ. (6.8) This and the continuity of the Lyapunov exponent (see, e.g., [5, 7, 33]) guarantees that(6.9) ω almost surely, for any δ > L large, one has − η • sup E ∈ I ρ ( E )(1 + δ ) L ≤ inf λ j ∈ I log a • j where η • is defined in Theorem 1.6.To use the analysis performed in section 3, we also need a description for the ( λ j ) j , i.e., the Dirichleteigenvalues of H ω,L . Therefore, we will use the results of [13], [22] and [21] (see also [14]).We first recall the Minami estimate satisfied by H ω,L (see, e.g., [8] and references therein): thereexists C > I ⊂ R , one has P (tr( I ( H ω,L ))) ≥ ≤ E (tr( I ( H ω,L ))[tr( I ( H ω,L )) − ≤ C | I | ( L + 1) . Here, I ( H ) denotes the spectral projector for the self-adjoint operator H onto the energy interval I .By a simple covering argument, this entails the following estimate P (cid:0) ∃ i = j s.t. | λ i − λ j | ≤ L − q (cid:1) ≤ CL − q +2 . Thus, for q > 3, a Borel-Cantelli argument yields, that(6.10) ω almost surely, for L sufficiently large, min i = j | λ i − λ j | ≥ L − q . The proofs of the main results in the random case. We are now going to prove theresults stated in section 1.3.6.2.1. The proof of Theorem 1.6. As for Theorem 1.2, this result follows from Theorem 3.1. Thepoint (1) is proved exactly as the point (1) in Theorem 1.2. Point (2) follows immediately fromTheorem 3.1 and (6.9). This completes the proof of Theorem 1.6.6.2.2. The proof of Theorem 1.7. Recall that κ ∈ (0 , N (see, e.g., [5, 7, 33]) implythat, ω almost surely, one has(6.11) { λ j ∈ I } L + 1 → Z I dN ( E ) . This, in particular, shows that, if I ⊂ ◦ Σ is a compact interval, then, ω almost surely, for L sufficientlylarge, I is covered by intervals of the form [ λ j , λ j +1 ] and their number is of size ≍ L (actually thisholds for λ j ∈ I + [ − ε, ε ] if ε > d j ≥ L − q (for any q > λ j ∈ I . Thus, Theorems 3.1, 3.2 and 3.3 and theestimate (6.8) guarantee that, ω almost surely, all the resonances in the strip I − i [ e − L κ , 0) aredescribed by Theorem 3.3. Indeed, for such a resonance the imaginary part must be larger than − e − L κ ; thus, by Theorem 3.1, for every rectangle [( λ j + λ j − ) / , ( λ j + λ j +1 ) / − i [ e − L κ , 0) containinga resonance, one has a j . e − L κ L q Thus, a j ≪ d j and one can apply Theorem 3.3 to compute theresonance.Let us count the number of those resonances. Therefore, let ℓ L = τ L κ where τ is to be chosen.By (6.8) and (6.10), ω almost surely, one has a j ≪ d j for all j such that λ j ∈ I as long as theDirichlet eigenvalue λ j is associated to a localization center in J , L − ℓ L K (actually it holds for λ j ∈ I + [ − ε, ε ] if ε > λ j ) j that are associated to a localization center in J , L − ℓ L K . By formula (3.19), each ofthese eigenvalues gives rise to a single simple resonance the imaginary part of which is of size ≍ a j ;it lies above the line { Im z ≥ e − ρℓ L = e − L κ } for τ ρ = 1. Actually, the estimate (6.10) guaranteesthat d j ≥ L − q (for any q > ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 65 above a line Im z ≥ − L − q . Moreover, by Lemma 6.2, we know there at most Cℓ L eigenvalues λ j that do not have their localization center in J , L − ℓ L K . Thus, we obtain, ω almost surely,lim L → + ∞ L (cid:8) z resonance of H ω,L s.t. Re z ∈ I, Im z ≥ − e − L κ (cid:9) = Z I dN ( E ) . Point (2) is proved in the same way. Pick λ ∈ (0 , E n ( E ) and Lyapunov exponent E ρ ( E ). Assume E isas in point (2). Then, ω almost surely, the reasoning done above shows that, for any η > 0, thereexists ε > ε ∈ (0 , ε ) and δ ∈ (0 , δ ), for L sufficiently large one has, λ l e.v of H N ω,L in E + ε n ( E ) [ − η, − η ] suchthat − e η • ρ ( E ) δL . e η • ρ ( E ) λ L a l . − e − η • ρ ( E ) δL ≤ (cid:8) z resonance of H • ω,L in R • ( E, λ, L, ε, δ ) (cid:9) ≤ λ l e.v of H N ω,L in E + ε n ( E ) [ − − η, η ] suchthat − e η • ρ ( E ) δL . e η • ρ ( E ) λ L a l . − e − η • ρ ( E ) δL Using Theorem 6.2 and the continuity of the Lyapunov exponent in conjunction with the definitionof a j (see (2.4) and (2.13)), we obtain that, ω almost surely, for any η > 0, there exists ε > ε ∈ (0 , ε ) and δ ∈ (0 , δ ), for L sufficiently large one has, e.v of H N ω,L in E + ε n ( E ) [ − η, − η ]with localization center in I • ( L, δ, − η ) ≤ (cid:8) z resonance of H • ω,L in R • ( E, λ, L, ε, δ ) (cid:9) ≤ e.v of H N ω,L in E + ε n ( E ) [ − − η, η ]with localization center in I • ( L, δ, η ) where I N ( L, λ, δ, η ) is the interval (here [ r ] denotes the integer part of r ∈ R ) I N ( L, λ, δ, η ) = [ Lλ ] + J − Lδ (1 + η ) , Lδ (1 + η ) K and, I Z ( L, λ, δ, η ) is the union of intervals I Z ( L, λ, δ, η ) = (cid:18)(cid:20) Lλ (cid:21) + J − Lδ (1 + η ) , Lδ (1 + η )) K (cid:19) ∪ (cid:18)(cid:20) L (cid:18) − λ (cid:19)(cid:21) + J − Lδ (1 + η )) , Lδ (1 + η )) K (cid:19) . Now, using the exponential localization of the eigenfunctions, one has that, ω almost surely, forany η > 0, there exists ε > ε ∈ (0 , ε ) and δ ∈ (0 , δ ), for L sufficiently large, onehas(6.12) (cid:26) e.v of H N ω,L,λ,δ, − η, • in E + ε n ( E ) [ − η, − η ] (cid:27) ≤ (cid:8) z resonance of H • ω,L in R • ( E, λ, L, ε, δ ) (cid:9) ≤ (cid:26) e.v of H N ω,L,λ,δ, η, • in E + ε n ( E ) [ − − η, η ] (cid:27) where H N ω,L,λ,δ,η, • = (cid:16) H N ω,L (cid:17) | I • ( L,λ,δ,η ) with Dirichlet boundary conditions at the edges of the interval I • ( L, λ, δ, η ).This immediately yields point (2) for λ ∈ (0 , 1) using (6.11) for the operators H N ω,L,λ,δ,η, • . The case λ = 1 is dealt with in the same way.As already said, point (3) is an “integrated” version of point (2). Using the same ideas as above,partitioning I = ∪ Pp =0 I p s.t. | I p | ∼ ε centered in E p , one proves P X p =0 (cid:26) e.v of H − ω,p,L, • in E p + ε n ( E p ) [ − η, − η ] (cid:27) ≤ (cid:8) z resonance of H • ω,L in I + (cid:2) − e − L κ , − e − cL (cid:3)(cid:9) ≤ P X p =0 (cid:26) e.v of H + ω,p,L, • in E p + ε n ( E p ) [ − − η, η ] (cid:27) where • H − ω,p,L, • is the operator H N ω restricted to – J L κ , (inf( cρ − ( E p ) , − η ) L K if • = N , – to J L κ , (inf( cρ − ( E p ) , / − η ) L K ∪ J (1 − inf( cρ − ( E p ) , / η ) L, L − L κ K if • = Z ; • H + ω,p,L, • is the operator H N ω restricted to – J L κ / , (inf( cρ − ( E p ) , 1) + η ) L K if • = N , – to J L κ / , (inf( cρ − ( E p ) , / η ) L K ∪ J (1 − inf ( cρ − ( E p ) , / − η ) L, L − L κ / K if • = Z ;In the computation above, we used the continuity of both, the density of states E n ( E ) andLyapunov exponent E ρ ( E ). Thus, we obtain (cid:8) z resonance of H • ω,L in I + (cid:0) −∞ , e − cL (cid:3)(cid:9) = L P X p =0 inf( cρ − ( E p ) , n ( E p ) | I p | + o (1) + (cid:8) z resonance of H • ω,L in I + (cid:0) −∞ , e − L κ (cid:3)(cid:9) . The last term being controlled by Theorem 1.10, one obtains point (3) as the Riemann sum in theright hand side above converges to the integral in the right hand side of (1.18) as ε → 0. Thiscompletes the proof of Theorem 1.7. (cid:3) The proof of Theorem 1.8. The proof of Theorem 1.8 relies on [13, Theorem 1.13] whichdescribes the local distribution of the eigenvalues and localization centers ( E j,ω , x j,ω ): namely, onehas(6.13) lim L → + ∞ P ω ; ( n ; E j,ω ∈ E + L − I x j,ω ∈ L C ) = k ... ... ( n ; E j,ω ∈ E + L − I p x j,ω ∈ L C p ) = k p = p Y n =1 e − ˜ µ n (˜ µ n ) k n k n !where ˜ µ n := n ( E ) | I n || C n | for 1 ≤ n ≤ p .Recall that ( z Lj ( ω )) j are the resonances of H ω,L . By the argument used in the proof of Theorem 1.7, ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 67 we know that, ω almost surely, all the resonances in K L := [ E − ε, E + ε ] + i (cid:2) − e − L κ , (cid:3) areconstructed from the ( λ • j , a • j ) by formula (3.19). Thus, up to renumbering, the rescaled real andimaginary parts (see (1.19)) become x j = (Re z • l,L ( ω ) − E ) L = ( λ j − E ) L + O ( La j ) = ( E j,ω − E ) L + O ( Le − L κ ) y j = − L log | Im z • l,L ( ω ) | = − log a • j L + O (1 /L ) = ρ ( E ) d • j,ω L + o (1) . where λ j = E j,ω and x j,ω is the associated localization center; here we used the continuity of E ρ ( E ).On the other hand, for the resonances below the line in { Im z = − e − L κ } , one has y j . L κ − . Soall these resonances are “pushed upwards” towards the upper half-plane. Hence, the statement ofTheorem 1.8 is an immediate consequence of (6.13). (cid:3) The proof of Theorem 1.9. Using the computations of the previous section, as E = E ′ ,Theorem 1.9 is a direct consequence of [22, Theorem 1.2] (see also [13, Theorem 1.11]).6.2.5. The proof of Theorem 1.10. Consider equations (2.4) and (2.8). By Theorem 6.2 andLemma 6.2, ω almost surely, for L large, the number of ( a • j ) j larger than e − ℓ L is bounded by Cℓ L . Solving (2.4) and (2.8) in the strip { Re E ∈ I, Im E < − e − ℓ L } , we can write S L ( E ) = S − L ( E ) + S + L ( E ) where S − L ( E ) := X a N j ≤ e − ℓL a N j λ j − E and S + L ( E ) := X a N j >e − ℓL a N j λ j − E and similarly decompose Γ L ( E ) = Γ − L ( E ) + Γ + L ( E ). For L large, one then has(6.14) sup Im E< − e − ℓL k S − L ( E ) k + k Γ − L ( E ) k ≤ e − ℓ L . (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) R~ λ j λ j+1 λ j−1 U U j j j new path Figure 8: The new pathThe count of the number of resonances given by the proofof Theorems 2.1 and 2.2 then shows that the equations (2.4)and (2.8) where S L and Γ L are respectively replaced by S + L and Γ + L have at most Cℓ L solutions in the lower half plane.The equations where S L and Γ L are replaced by S + L andΓ + L we will call the +-equations. The analogue of Theo-rems 3.1, 3.2 and 3.3 for the +-equations and Theorem 6.2show that the only solutions to the +-equations in the strip { Re E ∈ I, − e − ℓ L / < Im E < − e − ℓ L / } are given by formu-las (3.19) and (3.20) for the eigenvalues of the Dirichlet prob-lem associated to a localization center in J L − ℓ L , L − ℓ L / K if • = N and in J ℓ L / , ℓ L K ∪ J L − ℓ L , L − ℓ L / K if • = Z . Thus,these zeros are simple and separated by a distance at least L − from each other (recall (6.10)).Moreover, we can cover the interval I by intervals of the type [( λ j + λ j − ) / , ( λ j + λ j +1 ) / I ⊂ j + [ j = j − (cid:20) λ j + λ j − , λ j + λ j +1 (cid:21) where λ j − − I , λ j + I , λ j − ∈ I and λ j + ∈ I . Consider now the line { Im E = − e − ℓ L } and itsintersection with the vertical strip [( λ j + λ j − ) / , ( λ j + λ j +1 ) / − i R + . Three things may occur: (1) either e − ℓ L < a j dj | sin θ ( λ j ) | /C (the constant C is defined in Theorem 3.1), then, on theinterval [( λ j + λ j − ) / , ( λ j + λ j +1 ) / − ie − ℓ L , one has(6.16) (cid:12)(cid:12)(cid:12) S + L ( E ) + e − iθ ( E ) (cid:12)(cid:12)(cid:12) & (cid:12)(cid:12)(cid:12) det (cid:16) Γ + L ( E ) + e − iθ ( E ) (cid:17)(cid:12)(cid:12)(cid:12) & c > 0; recall that, on the interval I + ie − ℓ L , one has | sin θ ( E ) | & e − ℓ L > Ca j (the constant C is defined in Theorem 3.2), then, on the interval [( λ j + λ j − ) / , ( λ j + λ j +1 ) / − ie − ℓ L , one has again (6.16) for a possibly different constant; thisfollows from the proof of Theorem 3.2 (see in particular (3.15) and (3.16));(3) if we are neither in case (1) nor in case (2), then the line { Im E = − e − ℓ L } may cross R j (defined in Theorem 3.3; see also Fig. 7); we change the contour { Im E = − e − ℓ L } so as toenter ˜ U j until we reach the boundary of R j and then follow this boundary getting closer tothe real axis, turning around R j and finally reaching the line { Im E = − e − ℓ L } again on theother side of R j and following it up to the boundary of ˜ U j (see Figure 8); on this new line,the bound (6.16) again holds; moreover, this new line is closer to the real axis than the line { Im E = − e − ℓ L } .Let us call C ℓ the path obtained by gluing together the paths constructed in points (1)-(3) for j − ≤ j ≤ j + and the half-lines λ j − + λ j −− − i [ e − ℓ L , + ∞ ) and λ j + + λ j ++1 − i [ e − ℓ L , + ∞ ) (see (6.15)).One can then apply Rouch´e’s Theorem to compare the + equations to the equations (2.4) and (2.8):by (6.14) and (6.16), on the line C ℓ , one has (cid:12)(cid:12) S − L (cid:12)(cid:12) < (cid:12)(cid:12)(cid:12) S + L + e − iθ (cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12) det (cid:16) Γ L ( E ) + e − iθ ( E ) (cid:17) det (cid:16) Γ + L ( E ) + e − iθ ( E ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) det (cid:16) Γ L ( E ) + e − iθ ( E ) (cid:17)(cid:12)(cid:12)(cid:12) . Thus, the number of solutions to equations (2.4) and (2.8) below the line C ℓ is bounded by Cℓ L ;as C ℓ lies above { Im E = − e − ℓ L } , in the half-plane { Im E < − e − ℓ L } , the equations (2.4) and (2.8)have at most Cℓ L solutions. We have proved Theorem 1.10. (cid:3) The proof of Theorem 1.11. The first point in Theorem 1.11 is proved in the same way aspoint (2) in Theorem 1.7 up to the change of scales, L being replaced by ℓ L . Pick scales ( ℓ ′ L ) L satisfying (1.22) such that ℓ ′ L ≪ ℓ L . One has Lemma 6.4. Fix two sequences ( a L ) L and ( b L ) L such that a L < b L . With probability one, for L sufficiently large, n e.v. of H ω,ℓ L − ℓ ′ L /ρ in h a L + e − ℓ ′ L , b L − e − ℓ ′ L io ≤ { e.v. of H ω,L in [ a L , b L ] with loc. cent. in J , ℓ L K }≤ n e.v. of H ω,ℓ L +2 ℓ ′ L /ρ in h a L − e − ℓ ′ L , b L + e − ℓ ′ L io where ρ is given by Lemma 6.1.Proof. To prove Lemma 6.4, we apply Lemma 6.1 to L = ℓ L + ℓ ′ L (i.e. for the operator H ω restricted to the interval J , ℓ L + ℓ ′ L K ) and l L = ℓ ′ L . The probability of the bad set is the O ( L −∞ ),thus, summable in L . Using the localization estimate (6.3), one proves that • each eigenvalue of H ω,ℓ L − ℓ ′ L /ρ is at a distance of at most e − ℓ ′ L of an eigenvalue of H ω,L withloc. cent. in J , ℓ L K ; • each eigenvalue of H ω,L with loc. cent. in J , ℓ L K is at a distance of at most e − ℓ ′ L of aneigenvalue of H ω,ℓ L +2 ℓ ′ L /ρ . ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 69 Lemma 6.4 follows. (cid:3) The first point in Theorem 1.11 is then point (2) of Theorem 1.7 for the operator H ω,ℓ L − ℓ ′ L /ρ and H ω,ℓ L +2 ℓ ′ L /ρ and the fact that ℓ ′ L ≪ ℓ L .The proof of the second statement in Theorem 1.11 is very similar to that of Theorem 1.8. Fix I acompact interval in ◦ Σ. As ℓ L satisfies (1.22), one can find ℓ ′ L < ℓ ′′ L also satisfying (1.22) such that e − ℓ ′′ L ≪ e − ℓ L ≪ e − ℓ ′ L . For the same reasons as in the proof of Theorem 1.8, after rescaling, all theresonances in I − i ( −∞ , 0) outside the strip I − i h e − ℓ ′ L , e − ℓ ′′ L (cid:17) are then pushed to either 0 or i ∞ as L → + ∞ .On the other hand, the resonances in the strip I − i h e − ℓ ′ L , e − ℓ ′′ L (cid:17) are described by (3.19). Therescaled real and imaginary parts of the resonances (see (1.24)) now become x j = ( E j,ω − E ) ℓ L + o (1)and y j = ρ ( E ) d j,ω ℓ L + o (1).Now, to compute the limit of P ( { j ; x j ∈ I, y j ∈ J } = k ), using the exponential decay prop-erty (6.3), it suffices to use [13, Theorem 1.14]. Let us note here that [13, condition (1.50)] on thescales ( ℓ L ) L is slightly stronger than (1.22). That condition (1.22) suffices is a consequence of thestronger localization property known in the present case (compare Theorem 6.2 to [13, Assumption(Loc)]). This completes the proof of the second point in Theorem 1.11. The final statement in 1.11is proved in exactly the same way as Theorem 1.9.The proof of Theorem 1.11 is complete. (cid:3) The proofs of Proposition 1.3 and Theorem 1.12. Localization for the operator H N ω can bedescribed by the following Lemma 6.5. There exists ρ > and q > such that, ω almost surely, there exists C ω > s.t. for L sufficiently large, if (1) ϕ j,ω is a normalized eigenvector of H ω,L associated to E j,ω in Σ , (2) x j ( ω ) ∈ N is a maximum of x 7→ | ϕ j,ω ( x ) | in N ,then, for x ∈ N , one has (6.17) | ϕ j,ω ( x ) | ≤ C ω (1 + | x j ( ω ) | ) q/ e − ρ | x − x j ( ω ) | . Moreover, the mapping ω C ω is measurable and E ( C ω ) < + ∞ . This result for our model is a consequence of Theorem 6.1 (see, e.g., [28, 10, 7]) and [13, Theorem6.1].We thus obtain the following representation for the function Ξ ω (6.18) Ξ ω ( E ) = X j | ϕ j,ω (0) | E j,ω − E + e − i arccos( E/ As H N ω satisfies a Dirichlet boundary condition at − 1, one has(6.19) ∀ j, | ϕ j,ω (0) | > X j | ϕ j,ω (0) | = 1 . As E → − i ∞ , the representation (6.18) yieldsΞ ω ( E ) = − E − X j | ϕ j,ω (0) | E j,ω + O (cid:0) E − (cid:1) = − E − h δ , H N ω δ i + O (cid:0) E − (cid:1) = − ω E − + O (cid:0) E − (cid:1) . This proves the first point in Proposition 1.3.As a direct consequence of Theorem 6.1 and the computation leading to Theorem 5.2 (see sec-tion 5.1.2), we obtain that there exists ˜ c > L sufficiently large, with probability at least1 − e − ˜ cL , one has(6.20) sup Im E ≤− e − ˜ cL (cid:12)(cid:12)(cid:12)(cid:12)Z R dN ω ( λ ) λ − E − h δ , ( H ω,L − E ) − δ i (cid:12)(cid:12)(cid:12)(cid:12) ≤ e − ˜ cL . Taking(6.21) L = L ε ∼ c − | log ε | for some sufficiently small c > 0, this and Rouch´e’s Theorem implies that, with probability 1 − ε ,the number of zeros of Ξ ω (counted with multiplicity) in I + i ( −∞ , ε ] is bounded • from above by the number of resonances of H ω,L ε in I + ε + i ( −∞ , − ε − ε ]; • from below by the number of resonances of H ω,L ε in I − ε + i ( −∞ , − ε + ε ].where I + ε = [ a − ε, b + ε ] and I + ε = [ a + ε, b − ε ] if I = [ a, b ].Here, to apply Rouch´e’s Theorem, we apply the same strategy as in the proof of Theorem 1.10and construct a path bounding a region larger (resp. smaller) than I + ε + i ( −∞ , − ε − ε ] (resp. I − ε + i ( −∞ , − ε + ε ]) on which one can guarantee (cid:12)(cid:12)(cid:12) S L ( E ) + e − iθ ( E ) (cid:12)(cid:12)(cid:12) & c (see (6.21)) to be so small that c < min E ∈ I ρ ( E ). Applying point (3)of Theorem 1.7 to H ω,L ε with this constant c , we obtain that the number of resonances of H ω,L ε in I + ε + i ( −∞ , ε − ε ] (resp. I − ε + i ( −∞ , ε + ε ]) is upper bounded (resp lower bounded) by L ε Z I min (cid:18) cρ ( E ) , (cid:19) n ( E ) dE (1 + O (1)) = | log ε | c Z I cρ ( E ) n ( E ) dE (1 + O (1))= | log ε | Z I n ( E ) ρ ( E ) dE (1 + O (1)) . Hence, we obtain the second point of Proposition 1.3. The last point of this proposition is thenan immediate consequence of the arguments developed to obtain the second point if one takes intoaccount the following facts: • the minimal distance between the Dirichlet eigenvalues of H N ω,L is bounded from below by L − (see (6.10)), • the growth of the function E S L ( E ) + e − iθ ( E ) near the resonances (i.e. its zeros) is wellcontrolled by Proposition 3.1.Indeed, this implies that the resonances of H N ω,L are simple in I + i [ − e −√ L , 0) (one can choose largerrectangles) and that near each resonance one can apply Rouch´e’s Theorem to control the zero ofΞ ω . Note that this also yields ω -almost surely, there exists c ω such that(6.22) min z zero of Ξ ω z ∈ I + i ( − ε ω , inf Theorem 1.12 is a consequence of the following Theorem 6.4. There exists ˜ c > such that, ω almost surely, for L ≥ sufficiently large one has sup Re E ∈ I Im E< − e − ˜ cL (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ L,ω, ˜ ω ( E ) − Z R dN ˜ ω ( λ ) λ − E Z R dN ω ( λ ) λ − E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) S L,ω ( E ) − Z R dN ω ( λ ) λ − E (cid:12)(cid:12)(cid:12)(cid:12) ≤ e − ˜ cL ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 71 where Γ L,ω, ˜ ω ( E ) (resp. S L,ω ( E ) ) is the matrix Γ L ( E ) (resp. the function S L ( E ) ) (see (2.9) )constructed from the Dirichlet data on J , L K of − ∆ + V Z ω, ˜ ω,L (resp. − ∆ + V N ω,L ) (see (1.26) ) usingformula (2.9) (resp. (2.4) ). Theorem 6.4 is proved exactly as Theorem 5.2 except that one uses the localization estimates (6.2)instead of the Combes-Thomas estimates.Theorem 1.12 is then an immediate consequence of the estimate (6.20). Indeed, this implies thatif z is a resonance for e.g. H Nω,L in I + i (cid:0) −∞ , e ˜ cL (cid:3) , then | Ξ ω ( z ) | ≤ e − ˜ cL . By the last point ofProposition 1.3, ω almost surely, we know that the multiplicity of the zeros of Ξ ω is boundedby N ω . Moreover, for the zeros of Ξ ω in I + i ( − ε ω , z zero of Ξ ω z ∈ I + i ( − ε ω ,e − ˜ cL ) max | E − z | = e − ˜ cL (cid:12)(cid:12) Ξ ω ( E ) − (cid:0) S ω,L ( E ) + e − iθ ( E ) (cid:1)(cid:12)(cid:12) | Ξ ω ( E ) | < e − ˜ cL . This yields point (2) in Theorem 1.12 by an application of Rouch´e’s Theorem. Point (1) is obtainedin the same way using Proposition 3.1 that givesmax z resonance of H N ω,L z ∈ I + i ( − ε ω ,e − ˜ cL ) max | E − z | = e − ˜ cL (cid:12)(cid:12) Ξ ω ( E ) − (cid:0) S ω,L ( E ) + e − iθ ( E ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) S ω,L ( E ) + e − iθ ( E ) (cid:12)(cid:12) < e − ˜ cL . The case of H Z ω, ˜ ω,L is dealt with in the same way.This completes the proof of Theorem 1.12. (cid:3) Estimates on the growth of eigenfunctions. In the present section we are going to proveTheorems 6.2 and 6.3. At the end of the section, we also prove the simpler Lemma 6.2.The proof of Theorem 6.2 relies on locally uniform estimates on the rate of growth of the cocy-cle (1.15) attached to the Schr¨odinger operator that we present now. Define(6.23) T L ( E, ω ) = T ( E, ω L ) · · · T ( E, ω )where T ( E, ω j ) = (cid:18) E − ω j − 11 0 (cid:19) We start with an upper bound on the large deviations of the growth rate of the cocycle that isuniform in energy. Fix α > δ ∈ (0 , Lemma 6.6. Let I ⊂ R be a compact interval. For any δ > , there exists L δ > and η > suchthat, for L ≥ L δ and any K > , one has (6.24) P ∀ ≤ k ≤ K, ∀ E ∈ I, ∀k u k = 1 , log k T L ( E ; τ k ( ω )) u k L + 1 ≤ ρ ( E ) + δ ≥ − Ke − η ( L +1) where we recall that τ : Ω → Ω denotes the left shift (i.e. if ω = ( ω n ) n ≥ then [ τ ( ω )] n = ω n +1 for n ≥ ) and τ n = τ ◦ · · · ◦ τ n times. At the heart of this result is a large deviation principle for the growth rate of the cocycle (see [5,section I and Theorem 6.1]). As it also serves in the proof of Theorem 6.2, we recall it now. Onehas Lemma 6.7. Let I ⊂ R be a compact interval. For any δ > , there exists L δ > and η > suchthat, for L ≥ L δ , one has (6.25) sup E ∈ I k u k =1 P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) log k T L ( E ; ω ) u k L + 1 − ρ ( E ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ δ (cid:19) ≤ e − η ( L +1) . While this result is not stated as is in [5], it can be obtained from [5, Lemma 6.2 and Theorem6.1]. Indeed, by inspecting the proof of [5, Lemma 6.2 and Theorem 6.1], it is clear that thequantities involved (in particular, the principal eigenvalue of T ( z ; E ) = T ( z ) in [5, Theorem 4.3])are continuous functions of the energy E . Thus, taking this into account, the proof of [5, Theorem6.1] yields, for our cocycle, a convergence that is locally uniform in energy, that is, (6.25).To prove Theorem 6.2, in addition to Lemma 6.6, we also need to guarantee a uniform lower boundon the growth rate of the cocycle. We need this bound at least on the spectrum of H ω,L with agood probability. Actually, this is the best one can hope for: a uniform bound in the style of (6.24)will not hold.We prove Lemma 6.8. Fix I a compact interval and δ > . Pick u ∈ C s.t. k u k = 1 . For ≤ j ≤ L , if j ≤ L − , define K + j ( ω, L, δ, u ) := E ∈ I ; (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log (cid:13)(cid:13)(cid:13) T − L − ( j +1) ( E, τ j +1 ( ω )) u (cid:13)(cid:13)(cid:13) L − j − ρ ( E ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > δ and, if ≤ j , define K − j ( ω, L, δ, u ) := (cid:26) E ∈ I ; (cid:12)(cid:12)(cid:12)(cid:12) log k T j − ( E, ω ) u k j − ρ ( E ) (cid:12)(cid:12)(cid:12)(cid:12) > δ (cid:27) ; finally, define K + L ( ω, L, δ, u ) = ∅ = K − ( ω, L, δ, u ) .Recall that ( E j,ω ) ≤ j ≤ L are the eigenvalues of H ω,L and let x j,ω be the associated localization centers.For ≤ ℓ ≤ L , define the sets Ω + B ( L, ℓ, δ, u ) := ( ω ; ∃ j s.t. L − x j,ω ≥ ℓ and E j,ω ∈ K + x j,ω ( ω, L, δ, u ) ) and Ω − B ( L, ℓ, δ, u ) := ( ω ; ∃ j s.t. x j,ω ≥ ℓ and E j,ω ∈ K − x j,ω ( ω, L, δ, u ) ) . Then, the sets Ω ± B ( L, ℓ, δ, u ) are measurable and, for any δ > , there exists η > and ℓ > suchthat, for L ≥ ℓ ≥ ℓ , one has (6.26) max (cid:0) P (Ω + B ( L, ℓ, δ, u )) , P (Ω − B ( L, ℓ, δ, u )) (cid:1) ≤ ( L + 1) | I | e − η ( ℓ − − e − η . Here, the constant η is the one given by (6.25) . First, let us explain the meaning of Lemma 6.8. As, by Lemma 6.6, we already control the growthof the cocycle from above, we see that in the definitions of the set K − j ( ω, L, δ, u ) resp. K + j ( ω, L, δ, u ),it would have sufficed to require log k T j − ( E, ω ) u k j − ρ ( E ) ≤ − δ ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 73 resp. log (cid:13)(cid:13)(cid:13) T − L − ( j +1) ( E, τ j +1 ( ω )) u (cid:13)(cid:13)(cid:13) L − ( j + 1) − ρ ( E ) ≤ − δ. Hence, what Lemma 6.8 measures is that the probability that the cocycle at energy E n,ω leadingfrom a localization center x n,ω to either 0 or L decays at a rate smaller than the rate predicted bythe Lyapunov exponent.The sets Ω ± B ( L, ℓ, δ, u ) are the sets of bad configurations, i.e., the events when the rate of decayof the solution is far from the Lyapunov exponent. Indeed, for ω outside Ω ± B ( L, ℓ, δ ), i.e., if thereverse of the inequalities defining K ± j ( ω, L, δ, u ) hold, when j = x n,ω and E = E n,ω , then, weknow that the eigenfunction ϕ n,ω has to decay from the center of localization x n,ω (which is a localmaximum of its modulus) towards the edges of the intervals at a rate larger than γ ( E n,ω ) − δ . Theeigenfunction being normalized, at the localization center, it is of size at least L − / . This willentail the estimates (6.4) and (6.5) with a good probability.There is a major difference in the uniformity in energy obtained in Lemmas 6.8 and 6.6. InLemma 6.8, we do not get a lower bound on the decay rate that is uniform all over I : it is merelyuniform over the spectrum inside I (which is sufficient for our purpose as we shall see). Thereason for this difference in the uniformity between Lemma 6.6 and 6.8 is the same that makes theLyapunov exponent E ρ ( E ) in general only upper semi-continuous and not lower semi-continuous(in the present situation, it actually is continuous).We postpone the proofs of Lemmas 6.6 and 6.8 to the end of this section and turn to the proofs ofTheorems 6.2 and 6.3.6.3.1. The proof of Theorem 6.2. By Lemma 6.6, as T L ( E, ω ) ∈ SL (2 , R ), with probability at least1 − KLe − η ( L +1) , for L ≥ L δ and any K > 0, one also has ∀ ≤ k ≤ K, ∀ E ∈ I, ∀k u k = 1 , log k T − L ( E ; τ k ( ω )) u k L + 1 ≤ ρ ( E ) + δ. Now pick ℓ = C log L where C > P satisfying(6.27) P ≥ − L e − ηℓ , for L ≥ L δ and any l ∈ [ ℓ, L ] and any k ∈ [0 , L ], one also has(6.28) ∀ E ∈ I, ∀k u k = 1 , log k T − l ( E ; τ k ( ω )) u k l + 1 ≤ ρ ( E ) + δ. Let ϕ j,ω be a normalized eigenfunction associated to the eigenvalue E j,ω ∈ I with localization center x j,ω . By the definition of the localization center, one has(6.29) 1 L + 1 ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ϕ j,ω ( x j,ω ) ϕ j,ω ( x j,ω − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ L + 1 ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ϕ j,ω ( x j,ω + 1) ϕ j,ω ( x j,ω ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ . By the eigenvalue equation, for x ∈ J , L K , one has(6.30) (cid:18) ϕ j,ω ( x ) ϕ j,ω ( x − (cid:19) = T x − x j,ω ( E ; τ x j,ω ( ω )) ϕ j,ω ( x j,ω ) ϕ j,ω ( x j,ω − ! if x ≥ x j,ω ,T − x j,ω − x ( E ; τ x ( ω )) ϕ j,ω ( x j,ω ) ϕ j,ω ( x j,ω − ! if x ≤ x j,ω . Hence, by (6.24) and (6.28), with probability at least 1 − L e − ηℓ − L − p , if | x j,ω − x | ≥ ℓ , for x j,ω < x ≤ L , one has e − ( ρ ( E j,ω )+ δ ) | x − x j,ω | √ L + 1 ≤ e − ( ρ ( E j,ω )+ δ ) | x − x j,ω | (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ϕ j,ω ( x j,ω ) ϕ j,ω ( x j,ω − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) T x − x j,ω ( E ; τ x j,ω ( ω )) (cid:18) ϕ j,ω ( x j,ω ) ϕ j,ω ( x j,ω − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ϕ j,ω ( x ) ϕ j,ω ( x − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) (6.31)and, for 0 ≤ x < x j,ω , one has (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ϕ j,ω ( x ) ϕ j,ω ( x − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) T − x − x j,ω ( E ; τ x j,ω ( ω )) (cid:18) ϕ j,ω ( x j,ω ) ϕ j,ω ( x j,ω − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≥ e − ( ρ ( E j,ω )+ δ ) | x − x j,ω | (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ϕ j,ω ( x j,ω ) ϕ j,ω ( x j,ω − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≥ e − ( ρ ( E j,ω )+ δ ) | x − x j,ω | √ L + 1(6.32)On the other hand, by the definition of the Dirichlet boundary conditions, we know that (cid:18) ϕ j,ω (0) ϕ j,ω ( − (cid:19) = ϕ j,ω (0) (cid:18) (cid:19) and (cid:18) ϕ j,ω ( L + 1) ϕ j,ω ( L ) (cid:19) = ϕ j,ω ( L ) (cid:18) (cid:19) . Thus, ϕ j,ω (0) T x j,ω − ( E ; ω ) (cid:18) (cid:19) = (cid:18) ϕ j,ω ( x j,ω ) ϕ j,ω ( x j,ω − (cid:19) and ϕ j,ω ( L ) (cid:18) (cid:19) = T L − x j,ω − ( E ; τ x j,ω +1 ( ω )) (cid:18) ϕ j,ω ( x j,ω + 1) ϕ j,ω ( x j,ω ) (cid:19) . Thus, for ω Ω + B ( L, ℓ, δ, u + ) ∪ Ω − B ( L, ℓ, δ, u − ) where we have set u − := (cid:18) (cid:19) and u + := (cid:18) (cid:19) , weknow that e − ( ρ ( E j,ω ) − δ )( L − x j,ω ) ≤ (cid:13)(cid:13)(cid:13) T − L − x j,ω − ( E ; τ x j,ω +1 ( ω )) u + (cid:13)(cid:13)(cid:13) and e − ( ρ ( E j,ω ) − δ ) x j,ω ≤ (cid:13)(cid:13) T x j,ω − ( E ; ω ) u − (cid:13)(cid:13) Thus, we obtain that, for ω Ω + B ( L, ℓ, δ, u + ) ∪ Ω − B ( L, ℓ, δ, u − ), one has | ϕ j,ω ( L ) | = (cid:13)(cid:13)(cid:13)(cid:13) T − L − x j,ω ( E ; τ x j,ω +1 ( ω )) (cid:18) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ϕ j,ω ( x j,ω + 1) ϕ j,ω ( x j,ω ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ e − ( ρ ( E j,ω ) − δ )( L − x j,ω − (6.33)and | ϕ j,ω (0) | = (cid:13)(cid:13)(cid:13)(cid:13) T x j,ω ( E ; τ x j,ω ( ω )) (cid:18) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ϕ j,ω ( x j,ω ) ϕ j,ω ( x j,ω − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ e − ( ρ ( E j,ω ) − δ )( x j,ω − . (6.34)The estimates given by Lemma 6.8 on the probability of Ω + B ( L, ℓ, δ, u + ) and Ω − B ( L, ℓ, δ, u − ) for ℓ = C log L and the estimate (6.27) then imply that, with a probability at least 1 − L e − η ( ℓ − − L − p ,the bounds (6.31), (6.32), (6.33) and (6.34) hold. Thus, picking ℓ = C log L for C > η , thus, on δ and p ), these bounds hold with a probability at least 1 − L − p .This complete the proof of Theorem 6.2. (cid:3) ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 75 Remark 6.1. One may wonder whether the uniform growth estimate given by Lemmas 6.6 and 6.8is actually necessary in the proof of Theorem 6.2. That they are necessary is due to the fact thatboth the eigenvalue E j,ω and the localization center x j,ω (and, thus, the vector (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ϕ j,ω ( x j,ω ) ϕ j,ω ( x j,ω − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) )depend on ω . Thus, (6.25) is not sufficient to estimate the second term in the left hand sides of (6.31)and (6.32).6.3.2. The proof of Theorem 6.3. To prove Theorem 6.3, we follow the strategy that led to theproof of Theorem 6.2. First, note that (6.31) and (6.32) provide the expected lower bounds on theeigenfunction with the right probability. As for the upper bound, by (6.30), using the conclusionsof Theorem 6.2 and the bounds given by Lemma 6.6, we know that, e.g. for 0 ≤ x < x j,ω (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ϕ j,ω ( x ) ϕ j,ω ( x − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) T x ( E ; ω ) (cid:18) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) | ϕ j,ω (0) | ≤ e ( ρ ( E j,ω )+ δ ) x e − ( ρ ( E j,ω ) − δ ) x j,ω ≤ e − ( ρ ( E j,ω ) − Cδ ) | x − x j,ω | if (1 + C ) x ≤ ( C − x j,ω , i.e., 2(1 + C ) − x j,ω ≤ x j,ω − x .For x ≥ x j,ω one reasons similarly and, thus, completes the proof of Theorem 6.3. (cid:3) Remark 6.2. Actually, as the proof shows, the results one obtains are more precise than the claimsmade in Theorem 6.3 (see [24]).6.3.3. The proof of Lemma 6.8. The proofs for the two sets Ω ± B ( L, ℓ, δ, u ) are the same. We will onlywrite out the one for Ω + B ( L, ℓ, δ, u ). Let us first address the measurability issue for Ω + B ( L, ℓ, δ, u ).The functions ω E j,ω and ω ϕ j,ω are continuous (as the eigenvalues and eigenvectors of finitedimensional matrices depending continuously on the parameter ω = ( ω j ) ≤ j ≤ L ). Thus, for fixed j ,the sets { ω ; E j,ω ∈ K − j ( ω, L, δ, u ) } and { ω ; x j,ω > j } are open (we used the definition of x j,ω asthe left most localization center (see Theorem 6.2)). This yields the measurability of Ω + B ( L, ℓ, δ, u ).We claim that(6.35) 1 L + 1 Ω + B ( L,ℓ,δ,u ) ≤ L +1 − ℓ X j =0 h δ j , K + j ( ω,L,δ,u ) ( H ω,L ) δ j i where K + j ( ω,L,δ,u ) ( H ω,L ) denotes the spectral projector associated to H ω,L on the set K + j ( ω, L, δ, u ).Indeed, if one has E j,ω 6∈ K + x j,ω ( ω, L, δ, u ) for all j then the left hand side of (6.35) vanishes and theright hand side is non negative. On the other hand, if, for some j , one has 0 ≤ x j,ω ≤ L − ℓ and E j,ω ∈ K + x j,ω ( ω, L, δ, u ) then, we compute L − ℓ X l =0 h δ l , K + j ( ω,L,δ,u ) ( H ω,L ) δ l i = L − ℓ X l =0 X k s.t E k,ω ∈K + j ( ω,L,δ,u ) | ϕ k,ω ( l ) | ≥ | ϕ j,ω ( x j,ω ) | ≥ L + 1 ≥ L + 1 Ω + B ( L,ℓ,δ,u ) by the definition of x j,ω .An important fact is that, by construction (see Lemma 6.8), the set of energies K + j ( ω, L, δ, u ) doesnot depend on ω j . Hence, denoting by E ω j ( · ) the expectation with respect to ω j and E ˆ ω j ( · ) theexpectation with respect to ˆ ω j = ( ω k ) k = j , we compute E L − ℓ X j =0 h δ j , K + j ( ω,L,δ,u ) ( H ω,L ) δ j i = L − ℓ X j =0 E ˆ ω j (cid:16) E ω j (cid:16) h δ j , K + j ( ω,L,δ,u ) ( H ω,L ) δ j i (cid:17)(cid:17) As ω j is assumed to have a bounded compactly supported distribution and as K + j ( ω, L, δ, u ) doesnot depend on ω j , a standard spectral averaging lemma (see, e.g., [37, Theorem 11.8]) yields E ω j (cid:16) h δ j , K + j ( ω,L,δ,u ) ( H ω,L ) δ j i (cid:17) ≤ |K + j ( ω, L, δ, u ) | where | · | denotes the Lebesgue measure. Thus, we obtain(6.36) E L − ℓ X j =0 h δ j , K + j ( ω,L,δ,u ) ( H ω,L ) δ j i ≤ L − ℓ X j =0 E ˆ ω j (cid:16) |K + j ( ω, L, δ, u ) | (cid:17) = L − ℓ X j =0 E (cid:16) |K + j ( ω, L, δ, u ) | (cid:17) . By Lemma 6.7 and the Fubini-Tonelli theorem, we know that E (cid:16) |K + j ( ω, L, δ, u ) | (cid:17) = E (cid:18)Z I K + j ( ω,L,δ,u ) ( E ) dE (cid:19) = Z I E (cid:16) K + j ( ω,L,δ,u ) ( E ) (cid:17) dE ≤ | I | sup E ∈ I P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log (cid:13)(cid:13)(cid:13) T − L − ( j +1) ( E, ω ) u (cid:13)(cid:13)(cid:13) L − j − ρ ( E ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > δ ≤ | I | e − η ( L − j ) . Taking the expectation of both sides of (6.35) and plugging this into (6.36), we obtain P (Ω + B ( L, ℓ, δ, u )) ≤ ( L + 1) | I | e − η ( ℓ − L − ℓ X j =0 e − ηj ≤ ( L + 1) | I | e − η ( ℓ − − e − η . In the same way, one obtains P (Ω − B ( L, ℓ, δ, u )) ≤ ( L + 1) | I | e − η ( ℓ − − e − η . This completes the proof of Lemma 6.8. (cid:3) Remark 6.3. This proof can be seen as the analogue of the so-called Kotani trick for products offinitely many random matrices (see , e.g., [10]).6.3.4. The proof of Lemma 6.6. The basic idea of this proof is to use the estimate (6.25), inparticular, the exponentially small probability and some perturbation theory for the cocycles so asto obtain a uniform estimate.Let η be given by (6.25). Fix η ′ < η/ I = ∪ j ∈ J [ E j , E j +1 ] where e − η ′ ( L +1) / ≤ E j +1 − E j ≤ e − η ′ ( L +1) ;thus, J . e η ′ ( L +1) .We now want to estimate what happens for E ∈ [ E j , E j +1 ]. Therefore, using (1.15) and (cid:18) E − V ω ( n ) − 11 0 (cid:19) − (cid:18) E j − V ω ( n ) − 11 0 (cid:19) = ( E − E j )∆ T where ∆ T := (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) (cid:19)(cid:29) (cid:28)(cid:18) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) we compute(6.38) T L ( E, ω ) = T L ( E j , ω ) + L X l =1 ( E − E j ) l S l ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 77 where S l := X n 1) small to be fixedlater. Assume moreover that L is so that δ L ≥ L δ where L δ is defined in Lemma 6.7. Then, byLemma 6.7, for m ∈ { , · · · , l } , one has(1) either n m − n m − ≤ L δ ; then, one has (cid:13)(cid:13) T n m − n m − − ( E j , τ n m − ω ) (cid:13)(cid:13) ≤ e C ( n m − n m − ) ;(2) or n m − n m − ≥ L δ ; then, by (6.25), with probability at least equal to 1 − e − η ( n m − n m − ) / ,one has (cid:13)(cid:13) T n m − n m − − ( E j , τ n m − ω ) (cid:13)(cid:13) ≤ e ( n m − n m − )( ρ ( E j )+ δ ) . Define G n , ··· ,n l = { m ∈ { , · · · , l } ; n m − n m − ≥ L δ } and B n , ··· ,n l = { , · · · , l } \ G n , ··· ,n l . By definition, one has(6.42) X m ∈ B n , ··· ,nl ( n m − n m − ) ≤ lL δ and X m ∈ G n , ··· ,nl ( n m − n m − ) ≥ L − lL δ . For a fixed sequence n < n < · · · < n m , the random variables (cid:16) T n m ′ − n m ′− − ( E j , τ n m ′ ω ) (cid:17) ≤ m ′ ≤ m are independent. Hence, by (6.25), for a fixed ( m , · · · , m K ) ∈ G n , ··· ,n l , one has P (cid:18) inf ≤ k ≤ K (cid:13)(cid:13)(cid:13) T n mk − n mk − − ( E j , τ n mk ω ) (cid:13)(cid:13)(cid:13) ≥ e ( ρ ( E j )+ δ )( n mk − n mk − ) (cid:19) ≤ e − η P Kk =1 n mk − n mk − . Thus, for ε ∈ (0 , P ∃ ( m , · · · , m K ) ∈ G n , ··· ,n l s.t. K X k =1 n m k − n m k − ≥ εL inf ≤ k ≤ K (cid:13)(cid:13)(cid:13) T n mk − n mk − − ( E j , τ n mk − ω ) (cid:13)(cid:13)(cid:13) ≥ e ( ρ ( E j )+ δ )( n mk − n mk − ) ≤ L l e − ηεL . Hence, with probability at least 1 − L l e − ηεL , we know that ∃ ( m , · · · , m K ) ∈ G n , ··· ,n l s.t. K X k =1 n m k − n m k − ≥ L − lL δ − εL ∀ ≤ k ≤ K, (cid:13)(cid:13)(cid:13) T n mk − n mk − − ( E j , τ n mk − ω ) (cid:13)(cid:13)(cid:13) ≤ e ( ρ ( E j )+ δ )( n mk − n mk − ) . Using estimates (6.42) and (6.39) for the remaining terms in the product below, for any given m -uple ( n , · · · , n m ), one obtains P l Y m =1 (cid:13)(cid:13) T n m − n m − − ( E j , τ n mk − ω ) (cid:13)(cid:13) ≤ e ( ρ ( E j )+ δ )(1 − ε )( L − lL δ )+ C ( εL + lL δ ) ≥ − L l e − ηεL . Hence, with probability at least 1 − l L l e − ηεL , for 1 ≤ l ≤ l , we estimate k S l k ≤ X n 0, for L sufficiently large, withprobability at least 1 − e − ηεL/ , one has, for any E ∈ I , k T L ( E, ω ) k . e ( ρ ( E )+2 δ )( L +1) . ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 79 Hence, as T L ( E, ω ) ∈ SL (2 , R ), one has (cid:13)(cid:13) T − L ( E, ω ) (cid:13)(cid:13) . e ( ρ ( E )+2 δ )( L +1) .Using the fact that the probability measure on Ω is invariant under the shift (it is a productmeasure), we obtain (6.24). This completes the proof of Lemma 6.6. (cid:3) The proof of Lemma 6.2. Assume the realization ω is such that the conclusions of Lemma 6.1hold in I for the scales l L = 2 log L . Fix α > E L,ω be the set of indices of the eigenvalues( E j,ω ) ≤ j ≤ L of H ω,L having a localization center in J L − ℓ L , L K . Fix C > α > J L − Cℓ L , L K , i.e., Π C := J L − Cℓ L ,L K .Consider the following Gram matrices G ( E L,ω ) = (( h ϕ j,ω , ϕ j,ω i )) ( n,m ) ∈E L,ω ×E L,ω = Id N where N = E L,ω and G π ( E L,ω ) = (( h Π C ϕ j,ω , Π C ϕ j,ω i )) ( n,m ) ∈E L,ω ×E L,ω . By definition, the rank of G π ( E L,ω ) is bounded by the rank of Π C , i.e., by Cℓ L . Moreover, asby (6.3) one has k (1 − Π C ) ϕ j,ω k ≤ L q e − ρηCℓ L , one has k Id N − G π ( E L,ω ) k ≤ L q e − ρηCℓ L ≤ L q − Cρη . Thus, picking Cηρ > q + 2 yields that, for L sufficiently large, G π ( E L,ω ) is invertible and its rankis N . This yields E L,ω = N ≤ Cℓ L and the proof of Lemma 6.2 is complete. (cid:3) The half-line random perturbation: the proof of Theorem 1.13. Using the samenotations as in section 5.3, we can write H ∞ = (cid:18) H − ω, − | δ − ih δ || δ ih δ − | − ∆ +0 (cid:19) where • − ∆ +0 is the Dirichlet Laplacian on ℓ ( N ), • H − ω, − = − ∆ + V ω on ℓ ( { n ≤ − } ) with Dirichlet boundary conditions at 0.Define the operators Γ ω ( E ) := − ∆ +0 − E − h δ − | ( H − ω, − − E ) − | δ − i | δ ih δ | , ˜Γ ω ( E ) := H − ω, − − E − h δ | ( − ∆ +0 − E ) − | δ i | δ − ih δ − | . For Im E = 0, the numbers h δ − | ( H − ω, − − E ) − | δ − i and h δ | ( − ∆ +0 − E ) − | δ i have non vanishingimaginary parts of the same sign; hence, the complex number ( h δ − | ( H − ω, − − E ) − | δ − i ) − −h δ | ( − ∆ +0 − E ) − | δ i does not vanish. Thus, by rank one perturbation theory, (see, e.g., [37]), wethus know that Γ ω ( E ) and ˜Γ ω ( E ) are invertible for Im E = 0 and thatΓ − ω ( E ) = ( − ∆ +0 − E ) − + | ( − ∆ +0 − E ) − | δ ih δ | ( − ∆ +0 − E ) − | ( h δ − | ( H − ω, − − E ) − | δ − i ) − − h δ | ( − ∆ +0 − E ) − | δ i (6.46) ˜Γ − ω ( E ) = ( H − ω, − − E ) − + | ( H − ω, − − E ) − | δ − ih δ − | ( H − ω, − − E ) − | ( h δ | ( − ∆ +0 − E ) − | δ i ) − − h δ − | ( H − ω, − − E ) − | δ − i . (6.47)Thus, for Im E = 0, using Schur’s complement formula, we compute(6.48) ( H ∞ ω − E ) − = (cid:18) ˜Γ − ω ( E ) γ ( E ) γ ∗ (cid:0) E (cid:1) Γ − ω ( E ) (cid:19) . where γ ∗ (cid:0) E (cid:1) is the adjoint of γ (cid:0) E (cid:1) and γ ( E ) := −| ( H − ω, − − E ) − | δ − ih δ | Γ − ω ( E ) | The continuation through ( − , \ Σ . Let us start with the analytic continuation through( − , \ Σ.One easily checks that the function E 7→ h δ − | ( H − ω, − − E ) − | δ − i − is analytic outside Σ, theessential spectrum of H − ω, − and has simple zeros at the isolated eigenvalues of H − ω, − . Hence, E Γ − ω ( E ) can be analytically continued near an isolated eigenvalue of H − ω, − different from − − ω , using the spectral decomposition of of H − ω, − − E ) − , as for any eigenvector of H − ω, − ,say, ϕ , one has h δ − , ϕ i 6 = 0, for E , an isolated eigenvalue of H − ω, − different from − − ω near E , one checks that E ˜Γ − ω ( E ) can be analytically continuedto a neighborhood of E .Finally let us check what happens with γ . We compute γ ( E ) = −h δ − | ( H − ω, − − E ) − | δ − i − | ( H − ω, − − E ) − | δ − ih δ | ( − ∆ +0 − E ) − | . As E 7→ h δ − | ( H − ω, − − E ) − | δ − i − ( H − ω, − − E ) − is analytic near any isolated eigenvalue of ( H − ω, − ,we see that E γ ( E ) can be can be analytically continued to a neighborhood of an isolatedeigenvalue of H − ω, − .Hence, the representation (6.48) immediately shows that the resolvent ( H ∞ ω − E ) − can be continuedthrough ( − , \ Σ, the poles of the continuation being given by the zeros of the function E − h δ | ( − ∆ +0 − E ) − | δ ih δ − | ( H − ω, − − E ) − | δ − i = 1 − e iθ ( E ) Z R dN ω ( λ ) λ − E . No continuation through ( − , ∩ ◦ Σ . Let us study the analytic continuation through ( − , ∩ ◦ Σ. Considering the lower right coefficient of this matrix, we see that, when coming from upperhalf-plane through ( − , ∩ ◦ Σ, E ( H ∞ ω − E ) − can be continued meromorphically to the lowerhalf plane (as an operator from ℓ ( Z ) to ℓ ( Z )) only if E Γ − ω ( E ) can be meromorphically(as an operator from ℓ ( N ) to ℓ ( N )).As E ( − ∆ +0 − E ) − can be analytically continued (see section 2), by (6.46), the meromorphiccontinuation of E Γ − ω ( E ) will exist if and only if the complex valued map E g ω ( E ) := 1( h δ − | ( H − ω, − − E ) − | δ − i ) − − h δ | ( − ∆ +0 − E ) − | δ i can be meromorphically continued from the upper half-plane through ( − , ∩ ◦ Σ. Fix ω s.t. thespectrum of H − ω, − be equal to Σ and pure point (this is almost sure (see, e.g., [7, 33]). As δ − is acyclic vector for H − ω, − , for E an eigenvalue of H − ω, − , one then has(6.49) lim ε → + ( h δ − | ( H − ω, − − E − iε ) − | δ − i ) − = 0 . Hence, if the analytic continuation of g ω would exist, on ( − , ∩ ◦ Σ, it would be equal to(6.50) g ω ( E + i 0) = − h δ | ( − ∆ +0 − E − i − | δ i . By analyticity of both sides, this in turn would imply that (6.50) holds on the whole upper half-plane, thus, in view of the definition of g ω , that (6.49) holds on the whole upper half plane: this ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 81 is absurd! Thus, we have proved that, ω almost surely, E ( H ∞ ω − E ) − does not admit ameromorphic continuation through ( − , ∩ ◦ Σ.6.4.3. Absolutely continuity of the spectrum of H ∞ ω in ( − , ∩ ◦ Σ . Let us now prove that thespectral measure of H ∞ ω in ( − , ∩ ◦ Σ is purely absolutely continuous. Therefore, it suffices (see,e.g., [39, section 2.5] and [37, Theorem 11.6]) to prove that, for all E ∈ ( − , ∩ ◦ Σ, one haslim sup ε → + (cid:12)(cid:12) h δ , ( H ∞ ω − E − iε ) − δ i (cid:12)(cid:12) + (cid:12)(cid:12) h δ − , ( H ∞ ω − E − iε ) − δ − i (cid:12)(cid:12) < + ∞ . Using (6.46), (6.47) and (6.48), for Im E = 0, we compute(6.51) h δ − , ( H ∞ ω − E ) − δ − i = h δ − | ( H − ω, − − E ) − | δ − i − h δ | ( − ∆ +0 − E ) − | δ i · h δ − | ( H − ω, − − E ) − | δ − i , for n ≥ m ≤ h δ − n , ( H ∞ ω − E ) − δ m i = −h δ − n | ( H − ω, − − E ) − | δ − ih δ | ( − ∆ +0 − E ) − | δ m i − h δ | ( − ∆ +0 − E ) − | δ i · h δ − | ( H − ω, − − E ) − | δ − i and(6.53) h δ , ( H ∞ ω − E ) − δ i = h δ | ( − ∆ +0 − E ) − | δ i − h δ | ( − ∆ +0 − E ) − | δ i · h δ − | ( H − ω, − − E ) − | δ − i . Thus, to prove the absolute continuity of the spectral measure of H ∞ ω in ( − , ∩ ◦ Σ, it suffices toprove that, for E ∈ ( − , ∩ ◦ Σ, one haslim sup ε → + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h δ − | ( H − ω, − − E − iε ) − | δ − i ) − − h δ | ( − ∆ +0 − E − iε ) − | δ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h δ | ( − ∆ +0 − E − iε ) − | δ i ) − − h δ − | ( H − ω, − − E − iε ) − | δ − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! < ∞ . This is the case as • the signs of the imaginary parts of − ( h δ − | ( H − ω, − − E − iε ) − | δ − i ) − and h δ | ( − ∆ +0 − E − iε ) − | δ i are the same (negative if Im E < E > • for E ∈ ( − , h δ | ( − ∆ +0 − E − iε ) − | δ i has a finite limit when ε → + , • for E ∈ ( − , h δ | ( − ∆ +0 − E − iε ) − | δ i does not vanish in the limit ε → + .So, we have proved the part of Theorem 1.13 concerning the absence of analytic continuation ofthe resolvent of H ∞ ω through ( − , ∩ ◦ Σ and the nature of its spectrum in this set.6.4.4. The spectrum of H ∞ ω is pure point in ◦ Σ \ [ − , . Let us now prove the last part of The-orem 1.13. The proof relies again on (6.48). We pick β ∈ (0 , α/ 2) where α is determined byTheorem 6.1 for H − ω, − . Then, for n ≥ m ≤ 0, using the Cauchy-Schwartz inequality, for Im E = 0, we compute(6.54) E (cid:16)(cid:12)(cid:12) h δ − n , ( H ∞ ω − E ) − δ m i (cid:12)(cid:12) β (cid:17) ≤ |h δ | ( − ∆ +0 − E ) − | δ m i| · E (cid:18)(cid:12)(cid:12)(cid:12) h δ − n | ( H − ω, − − E ) − | δ − i (cid:12)(cid:12)(cid:12) β (cid:19) · E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − h δ | ( − ∆ +0 − E ) − | δ i · h δ − | ( H − ω, − − E ) − | δ − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β For J ⊂ ( − , \ Σ a compact interval, we know that, for n ≥ m ≤ • sup Im E =0 |h δ | ( − ∆ +0 − E ) − | δ m i| . e − cm by the Combes-Thomas estimates; • sup Im E =0 E (cid:18)(cid:12)(cid:12)(cid:12) h δ − n | ( H − ω, − − E ) − | δ − i (cid:12)(cid:12)(cid:12) β (cid:19) . e − βρn by the characterization (6.1) of localiza-tion in Σ for H − ω, − .It suffices now to estimate the last term in (6.54) using a standard decomposition of rank oneperturbations (see, e.g., [37, 2]), one writes11 − h δ | ( − ∆ +0 − E ) − | δ i · h δ − | ( H − ω, − − E ) − | δ − i = ω − − bω − − a where a and b only depend on ( ω − n ) n ≥ . Thus, as ( ω − n ) n ≥ have a bounded density, for Im E = 0,one has E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − h δ | ( − ∆ +0 − E ) − | δ i · h δ − | ( H − ω, − − E ) − | δ − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β ≤ E ( ω − n ) n ≥ E ω − (cid:12)(cid:12)(cid:12)(cid:12) ω − − bω − − a (cid:12)(cid:12)(cid:12)(cid:12) β ! ≤ C β < + ∞ . Thus, we have proved that, for J ⊂ Σ \ [ − , 2] a compact interval, for β ∈ (0 , α/ 2) and some ˜ ρ > n ≥ m ≤ 0, one hassup Im E =0Re E ∈ I E (cid:16)(cid:12)(cid:12) h δ − n , ( H ∞ ω − E ) − δ m i (cid:12)(cid:12) β (cid:17) < C β e − ˜ ρ ( m − n ) . In the same way, using (6.51) and (6.53), one proves thatsup Im E =0Re E ∈ I E (cid:16)(cid:12)(cid:12) h δ , ( H ∞ ω − E ) − δ i (cid:12)(cid:12) β + (cid:12)(cid:12) h δ − , ( H ∞ ω − E ) − δ − i (cid:12)(cid:12) β (cid:17) < + ∞ Thus, we have proved that, for some ˜ ρ > 0, one hassup Im E =0Re E ∈ I sup m ∈ Z E X n ∈ Z e ˜ ρ ( m − n ) (cid:12)(cid:12) h δ − n , ( H ∞ ω − E ) − δ m i (cid:12)(cid:12) β ! < + ∞ . Hence, we know that the spectrum of H ∞ ω in Σ \ [ − , 2] (as J can be taken arbitrary contained inthis set) is pure point associated to exponentially decaying eigenfunctions (see, e.g., [2, 1, 3]). Thiscompletes the proof of Theorem 1.13. ESONANCES FOR LARGE ONE-DIMENSIONAL “ERGODIC” SYSTEMS 83 Appendix In this section we study the eigenvalues and eigenvectors of H L (see Remark 1.4) near an energy E ′ that is an eigenvalue of both H +0 and H − k (see the ends of sections 4.1.3 and 4.1.4). We keepthe notations of sections 4.1.3 and 4.1.4.Let ϕ + ∈ ℓ ( N ) (resp. ϕ − ∈ ℓ ( Z − )) be normalized eigenvectors of H +0 (resp. H − k ) associated to E − . Thus, by (4.28) and (4.32), we can pick, for n ≥ l ∈ { , · · · , p − } ,(7.1) ϕ + np + l = ca l ( E ′ ) ρ n ( E ′ ) and ϕ −− np − l = c − b l ( E ′ ) ρ n ( E ′ ) . Assume L = N p + k and, for l ∈ { , · · · , L } , define ϕ ± ,L ∈ ℓ ( J , L K ) by(7.2) ϕ + ,Ll := ϕ + l , ϕ + ,L − = ϕ + ,LL +1 := ϕ + − = 0 and ϕ − ,Ll := ϕ − l − L , ϕ − ,L − = ϕ − ,LL +1 := ϕ − = 0 . Thus, one has(7.3) H L ϕ + ,L = E ′ ϕ + ,L + ϕ + L +1 δ L , H L ϕ − ,L = E ′ ϕ − ,L + ϕ −− L − δ and h ϕ + ,L , ϕ − ,L i = O ( N ρ N ( E )) . Recall that a k ( E ′ ) = 0 = b k ( E ′ ) (see sections 4.1.3 and 4.1.4); thus, by (7.1), one has(7.4) | ϕ −− L − | ≍ | ρ ( E ′ ) | n ≍ | ϕ + L +1 | . Moreover, as H L converges to H +0 in strong resolvent sense, for ε > L sufficiently large, H L has no spectrum in the compact E ′ + [ − ε, ε/ ∪ [ ε/ , ε ]. Let Π L be thespectral projector onto the interval [ ε/ , ε/ 2] that is Π L := 12 iπ Z | z − E ′ | = ε ( H L − z ) − dz . By (7.3),one computes (1 − Π L ) ϕ + ,L = ϕ + L +1 iπ Z | z − E ′ | = ε ( E ′ − z ) − ( H L − z ) − δ dz Thus, one gets(7.5) k (1 − Π L ) ϕ + ,L k + k (1 − Π L ) ϕ − ,L k . | ρ ( E ′ ) | N . Define ˜ χ + ,L = 1 k Π L ϕ + ,L k Π L ϕ + ,L and ˜ χ − ,L = 1 k Π L ϕ − ,L k Π L ϕ − ,L . The Gram matrix of ( ˜ χ + ,L , ˜ χ − ,L ) then reads Id+ O ( N ρ N ( E )). Orthonormalizing ( ˜ χ + ,L , ˜ χ − ,L ) into( χ + ,L , χ − ,L ) and, computing the matrix elements of Π L ( H L − E ′ ) in this basis, we obtain (cid:18) ϕ + L +1 h δ L , ϕ + ,L i ϕ + L +1 h δ , ϕ + ,L i ϕ −− L − h δ L , ϕ − ,L i ϕ −− L − h δ , ϕ − ,L i (cid:19) + O ( N ρ N ( E )) = α ρ N ( E ) (cid:18) (cid:19) + O ( N ρ N ( E ))Thus, we obtain that the eigenvalues of H L near E ′ are given by E ′ ± αρ N ( E ) + O ( N ρ N ( E )) andthe eigenvectors by √ ( ϕ + ,L ± ϕ − ,L ) + O ( ρ N ( E )). 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