A note on fractional moments for the one-dimensional continuum Anderson model
aa r X i v : . [ m a t h - ph ] J u l A NOTE ON FRACTIONAL MOMENTS FOR THE ONE-DIMENSIONALCONTINUUM ANDERSON MODEL
EMAN HAMZA, ROBERT SIMS AND G ¨UNTER STOLZ
Abstract.
We give a proof of dynamical localization in the form of exponential decay of spatialcorrelations in the time evolution for the one-dimensional continuum Anderson model via the fractionalmoments method. This follows via exponential decay of fractional moments of the Green function,which is shown to hold at arbitrary energy and for any single-site distribution with bounded, compactlysupported density. Introduction
The fractional moment method (FMM) was initially developed for the discrete Anderson model in [3].It has recently been extended in [1] and [6] to cover continuum Anderson models, where it was shownthat, in any dimension d ≥
1, exponential decay of fractional moments of the Green function, e.g. (5)below, implies dynamical and spectral localization. In fact, as discussed below, the result on dynamicallocalization which is obtained via the FMM is stronger than what is obtained by other methods. Thefractional moment condition (5) has also been found to be a technically useful tool in other contexts,for example in the proof of Poisson statistics of eigenvalues of the Anderson model in finite volume [18]or vanishing of the d. c. electrical conductivity of an electron gas [2].The main goal of this note is to fill a gap in the literature, which is to show that the FMM appliesto one-dimensional continuum Anderson models. While localization properties of the one-dimensionalAnderson model are well understood via other methods, given the mentioned applications it is usefulto know that a proof via fractional moments can be given. In dimension d = 1 localization should holdin the Anderson model at all energies, independent of the disorder strength. To conclude this via theFMM, exponential decay of the fractional moments needs to be verified at all energies. For the discreteAnderson model this was done in the Appendix of [18].Here we will do this for the continuum one-dimensional Anderson model, which is a random operatorin L ( R ) of the form(1) H = H ( ω ) = − d dx + W + V ω . The background potential, W , is bounded, real-valued and 1-periodic, i.e. W ( x + 1) = W ( x ). Therandom potential is given by(2) V ω = X n ∈ Z η n ( ω ) f n , where we will assume that the single site potentials f n are translates f n ( x ) = f ( x − n ) of a non-negativeand bounded function f . Moreover, we suppose that f is supported on [0 , J of [0 , C ≥ c > cχ J ≤ f ≤ Cχ [0 , . For the random variables η n , we assume that they are independent and identically distributed. We willalso assume that their common distribution µ ( A ) = P ( η n ∈ A ) has a bounded density ρ with compactsupport, i.e.(4) k ρ k ∞ < ∞ , supp( ρ ) ⊂ [ η min , η max ] . Date : November 5, 2018.
Given any bounded interval Λ, we will denote by H Λ = H Λ ( ω ) the restriction of H to L (Λ) withDirichlet boundary conditions. By G Λ ( z ) = ( H Λ − z ) − we denote the resolvent of H Λ . We write χ x for the characteristic function of the interval [ x, x + 1]. By k · k we will denote Hilbert-Schmidt norm.Our main result is Theorem 1.1.
For any E ∈ R there exists a number s ∈ (0 , such that for all < s ≤ s there are η > and C < ∞ such that (5) E (cid:0) k χ x G Λ ( E ) χ y k s (cid:1) ≤ C e − η | x − y | , holds for every interval Λ with integer endpoints, all integers x, y ∈ Λ and E ∈ ( −∞ , E ] . Theorem 1.1 will be proven in Section 3. As a preparation we will show in Section 2 that for thecontinuum Anderson model given by (1) and (2) Furstenberg’s Theorem applies at all energies andthus, in particular, the Lyapunov exponent is positive at all energies. We show this under the weakerassumption that the distribution of the random coupling constants η n has non-discrete support bycombining results of [16] and [10].Theorem 1.1 implies dynamical and spectral localization at all energies: Theorem 1.2.
For any E ∈ R there exist η > and C < ∞ such that (6) E (sup k χ x g ( H ) P E ( H ) χ y k ) ≤ Ce − µ | x − y | for all integers x and y . Here the supremum is taken over all Borel measurable functions g which satisfy | g | ≤ pointwise and P E ( H ) is the spectral projection for H onto ( −∞ , E ] .Also, H almost surely has pure point spectrum with exponentially decaying eigenfunctions. An argument which shows that Theorem 1.2 follows from Theorem 1.1 was provided in [1]. However,to allow single-site potentials of small support as in (3) the proof in [1] needs to be slightly modified.We indicate the changes at the end of Section 3The particular choice g ( x ) = e itx , t ∈ R arbitrary, shows that (6) is a result on dynamical localization.The exponential decay bound on the right hand side is stronger than what has been obtained with othermethods. Note, however, that for the discrete one-dimensional Anderson model the analog of (6) wasalready obtained in [17] by a method which has not yet been extended to the continuum, however, see[12]. Spectral localization for H is, of course, not new, see e.g. [10] for a more general result. We includeit here for completeness and because it was shown in [1] how it follows by an argument using the RAGEtheorem from dynamical localization and thus, via Theorem 1.2, is a consequence of (5).As mentioned above, the discrete analog of our main result is proven in an appendix of [18]. Forcompleteness, we include an alternate proof of this fact in Section 4, where we use methods similar tothe ones in our proof of Theorem 1.1. There we will also include a new proof of boundedness of thefractional moments of Green’s function for the discrete Anderson model. For the ”off-diagonal case”, x = y in (37), this slightly streamlines earlier arguments, e.g. [3, 13], by using a change-of-variablesargument which was developed for the continuum FMM in [1]. A similar strategy was used in thecontext of unitary Anderson models in [14].Our proof of Theorem 1.1 in Section 3 uses Pr¨ufer variables which require to work at real energy E .The finite volume resolvent G Λ ( E ) is almost surely well defined as H Λ has discrete eigenvalues whichare strictly monotone in all the random parameters. For some applications and to also have a result forinfinite volume it is of interest to be able to extend our main result to complex energy, i.e. to considerenergies E + iε in Theorem 1.1 and its discrete analog Theorem 4.1 with bounds which are uniform in ε >
0. As discussed in Section 5, this can easily be done for Theorem 4.1. While we expect the sameto hold for the continuum, it does not seem to follow with our method of proof.In order to make our presentation self-contained, we will provide a variety of facts, well-known tothose familiar with a-priori solution bounds and the Pr¨ufer formalism, in an Appendix.
NOTE ON FRACTIONAL MOMENTS FOR THE ONE-DIMENSIONAL CONTINUUM ANDERSON MODEL 3 Furstenberg at all energies
In this section we consider the continuous one-dimensional Anderson model defined by (1) and (2)under the weaker assumption that the coupling constants have non-discrete distribution, i.e.(7) supp µ is not discrete . For fixed E ∈ R , let T ( η, E ) be the transfer matrices of − u ′′ + W u + ηf u = Eu from 0 to 1 and G ( E )the Furstenberg group to energy E , i.e. the closed subgroup of SL (2 , R ) generated by the matrices T ( η, E ) with η varying in the support of the single site distribution µ .The goal of this section is to prove the following result, which is optimal with respect to the use ofassumption (7) and thus of some interest by itself. Theorem 2.1.
For the continuum one-dimensional Anderson model given by (1), (2) and (7), theFurstenberg group G ( E ) is non-compact and strongly irreducible for all E ∈ R . For the definition of strong irreducibility see [5]. By Furstenberg’s Theorem [5], the above resultimplies that the Lyapunov exponent associated with G ( E ) is positive for all energies E ∈ R . That µ has non-discrete support is crucial here. Examples have been constructed showing that non-trivial butdiscretely supported single site distributions can lead to a discrete set of critical energies where G ( E )is compact or not strongly irreducible (and the Lyapunov exponent may vanish), see [11] or Section 5of [9].Theorem 2.1 follows from applying a slight generalization of the main result in [16], see Theorem 2.2below, to the methods developed in [10]. For the sake of completeness, we outline this argument.We begin by stating a generalization of the result in [16]. Let Q : R → R be locally integrable andfor j = 0 ,
1, take u j : R → C to be solutions of(8) − u ′′ j + Qu j = 0 , neither of which are identically zero. For any V : R → R with V ∈ L ( R ) and support contained in[0 , u ( λ ) the solution of(9) − u ′′ + ( Q + λV ) u = 0which satisfies u ( λ ) ( x ) = u ( x ) for all x <
0. Here we may consider coupling constants λ ∈ C . Thequestion of interest in this context is: Given a non-trivial function V , for how many values of λ is itpossible that the solution u ( λ ) , which for x < u , is proportional to u for x >
1? Thecase where u = u is discussed in [16]. Following their arguments, we define the Wronskian(10) b ( λ ) = W (cid:2) u , u ( λ ) (cid:3) ( x ) = u ( x ) u ′ ( λ ) ( x ) − u ′ ( x ) u ( λ ) ( x )for x >
1. The λ -set in question is given by the zeros of b . Theorem 2.2. If V is not identically zero and either (11) u = u (and possibly complex-valued)or (12) u and u are real-valued , then the zeros of b form a discrete set. In [16] this result is stated and proven for the case u = u . However, for the case of real-valuedsolutions u and u , the proof provided in [16] goes through without change if u = u . We will useboth versions of this result below. Proof. (of Theorem 2.1) Fix E ∈ R . Let D ( E ) = Tr [ T (0 , E )] denote the discriminant of − d /dx + W .The first step in our proof demonstrates that, without loss of generality, we may assume both 0 ∈ supp( µ ) and D ( E ) / ∈ {− , , } . This is easily seen by adjusting the periodic background V per . In EMAN HAMZA, ROBERT SIMS AND G¨UNTER STOLZ fact, let η be an accumulation point for supp( µ ). Consider ˜ D ( E ) = Tr [ T ( η , E )], the discriminant of − d dx + ˜ W where(13) ˜ W = W + η X n ∈ Z f ( · − n ) . Clearly(14) H ω = ˜ W + X n ∈ Z ˜ η n ( ω ) f ( · − n ) , where the random variables { ˜ η n } have distribution ˜ µ defined by ˜ µ ( M ) = µ ( M + η ), i.e. 0 ∈ supp(˜ µ ). If˜ D ( E ) / ∈ {− , , } , then we have completed the first step of this proof. If ˜ D ( E ) ∈ {− , , } , then E is an eigenvalue of an operator with quasi-periodic boundary conditions. To see this, define the familyof self-adjoint operators(15) H λ,θ = − d dx + ˜ W + λf on [0 , u (1) = e iθ u (0) and u ′ (1) = e iθ u ′ (0). It is clear that E is an eigenvalue of H λ,θ if and only if the corresponding discriminant Tr [ T ( η + λ, E )] is 2 cos( θ ). We conclude that if˜ D ( E ) = Tr [ T ( η , E )] ∈ {− , , } , then E is an eigenvalue of H ,π , H , π , or H , respectively. Since f ≥ f = 0, analytic perturbation theory, see e.g. [15], implies that there exists δ > λ ∈ ( − δ, δ ) \ { } , E is not an eigenvalue of H λ,π , H λ, π , and H λ, . This uses that all the eigenvaluesof H λ,θ are analytic and strictly increasing in λ , the latter being due to the Feynman-Hellmann formulawhich shows that (3) suffices to get positivity of the λ -derivative of eigenvalues.As η was an accumulation point, there exists λ ∈ ( − δ, δ ) \ { } such that η = η + λ ∈ supp( µ ).Defining ˜˜ W analogously to (13) with η replaced by η , we have completed step 1.Step 2 of this proof demonstrates the validity of Theorem 2.1 in the event that D ( E ) ∈ ( − , \ { } ,i.e. E is in a band of − d /dx + W without being at the “band center”. Let φ ± denote the linearlyindependent Floquet solutions of − φ ′′ + W φ = E φ , see e.g. [10] for details. Denote by u ( η ) the solutionof(16) − u ′′ + ( W + ηf ) u = E u which satisfies(17) u ( η ) ( x ) = (cid:26) φ + ( x ) for x < ,a ( η ) φ + ( x ) + b ( η ) φ − ( x ) for x > . A simple Wronskian argument shows that a ( η ) = 0 for all η , and by Theorem 2.2 (under condition(11)), the set { η ∈ C : b ( η ) = 0 } is discrete. Since the support of µ is not discrete, there exists a η ∈ supp( µ ) \ { } for which b ( η ) = 0. It is shown in [10] that G ( E ) contains a subgroup which isconjugate to the group generated by the matrices(18) Q − (cid:18) ρ ρ (cid:19) Q and Q − (cid:18) a ( η ) b ( η ) b ( η ) a ( η ) (cid:19) Q where Q = 12 (cid:18) − i i (cid:19) , and the numbers ρ and ρ are the Floquet multipliers, i.e. the eigenvalues of the transfer matrix T (0 , E ). D ( E ) ∈ ( − , \ { } means that ρ = e iω with ω ∈ (0 , π ) \ { π/ } . Using this and the explicit form ofthis group, it was shown to be non-compact and strongly irreducible in [10]. The same readily followsfor G ( E ).Step 3 finishes the proof in the case that | D ( E ) | >
2, i.e. E is in a gap of − d /dx + W . In thiscase, there exist real-valued linearly independent solutions u ± , each not identically zero, of(19) − u ′′ + W u = E u with u ± in L near ±∞ . Similar to above, we denote by u ± ( η ) the solution of(20) − u ′′ + ( W + ηf ) u = E u NOTE ON FRACTIONAL MOMENTS FOR THE ONE-DIMENSIONAL CONTINUUM ANDERSON MODEL 5 which satisfies(21) u ± ( η ) ( x ) = (cid:26) u ± ( x ) for x < ,a ± ( η ) u ± ( x ) + b ± ( η ) u ∓ ( x ) for x > . Using Theorem 2.2 (under condition (12)) for each of the four pairs ( u ± , u ± ), one finds that the set(22) { η ∈ C : a + ( η ) b + ( η ) a − ( η ) b − ( η ) = 0 } is discrete. Picking η ∈ supp( µ ) \ { } for which a + ( η ) b + ( η ) a − ( η ) b − ( η ) = 0, we will prove thatthe subgroup generated by T (0 , E ) and T ( η , E ) is non-compact and strongly irreducible repeatingarguments from [10].Since | D ( E ) | > T (0 , E ) has eigenvalues ρ and ρ − with ρ > ρ < −
1. Denote by(23) v ± = (cid:18) u ∓ (0) u ′∓ (0) (cid:19) the eigenvectors of T (0 , E ) corresponding to ρ and ρ − , respectively. Clearly, w n = T (0 , E ) n v + isunbounded, and therefore, the subgroup generated by T (0 , E ) alone is non-compact. As we haveshown that this group is non-compact, to prove that it is also strongly irreducible, we need only showthat each direction is mapped onto at least three distinct directions by this group, see e.g. [5]. First,suppose v is not in the direction of v + or v − . Then, the sequence w n = T (0 , E ) n v produces arbitrarilymany directions (as w n approaches the stable manifold generated by v − ). If v is in the direction of v + or v − , then T ( η , E ) v is not as a + ( η ) b + ( η ) a − ( η ) b − ( η ) = 0. By our previous argument then,˜ w n = T (0 , E ) n T ( η , E ) v produces arbitrarily many directions. This completes step 3 and the proofof Theorem 2.1. (cid:3) Proof of Theorem 1.1
Non-compactness and strong irreducibility of the Furstenberg group G ( E ), if known for all energiesin an interval, leads to consequences which go beyond positivity of the Lyapunov exponents. To statethe result which we need, denote by T ( n, k, E ) = T ω ( n, k, E ) the transfer matrix of H at energy E from k to n , i.e. the 2 × T ( n, k, E ) (cid:18) u ( k ) u ′ ( k ) (cid:19) = (cid:18) u ( n ) u ′ ( n ) (cid:19) for all solutions of − u ′′ + ( W + V ω ) u = Eu . Lemma 3.1.
Let I ⊂ R be a compact interval such that G ( E ) is non-compact and strongly irreduciblefor every E ∈ I . Then there exist α > , δ > and n ∈ N such that for all E ∈ I , n ≥ n and x ∈ R normalized, E ( k T ( n, , E ) x k − δ ) ≤ e − α n . This is essentially Lemma 5.2 of [10]. While the latter is stated in a more concrete setting, the aboveslightly abstracted version is what one gets from the argument provided in [10] to which we refer forthe proof.Thus, under the assumptions of Theorem 1.1, we conclude from Theorem 2.1 that Lemma 3.1 appliesto every compact interval I . To prove Theorem 1.1 it suffices to consider energies E ∈ I := [ E , E ],where E is a deterministic and strict lower bound of the potential W + V ω (which exists by ourassumptions). For energies below E exponential decay of the right hand side of (5) is a deterministicconsequence of Combes-Thomas bounds, e.g. [19].Our main tools in reducing (5) to Lemma 3.1 are the Pr¨ufer amplitudes and phases corresponding tosolutions of H Λ u = Eu . We introduce these as follows. Write Λ = [ a, b ] for integers a , b . For any E ∈ R , c ∈ [ a, b ] and θ ∈ R we denote by u c ( x, E, θ ) the solution of − u ′′ + ( W + V ω ) u = Eu which satisfies u ( c ) = sin θ and u ′ ( c ) = cos θ . By regarding this solution and its derivative in polar coordinates, wedefine the Pr¨ufer amplitude, R c ( x, E, θ ), and the Pr¨ufer phase, φ c ( x, E, θ ) , by writing(24) u c ( x, E, θ ) = R c ( x, E, θ ) sin φ c ( x, E, θ ) and u ′ c ( x, E, θ ) = R c ( x, E, θ ) cos φ c ( x, E, θ ) . EMAN HAMZA, ROBERT SIMS AND G¨UNTER STOLZ
For fixed E , we declare φ c ( c, E, θ ) = θ and require continuity of φ in x . In this manner we defineuniquely the functions R c ( x, E, θ ) and φ c ( x, E, θ ) which are jointly continuous in x and E .For the remainder of this section, finite positive constants which can be chosen uniform in the givencontext may change their value from line to line. Proof. (of Theorem 1.1) We may assume that the integers x , y satisfy x ≤ y (if x > y use that k χ x G Λ ( E ) χ y k = k ( χ x G Λ ( E ) χ y ) ∗ k = k χ y G Λ ( E ) χ x k ). Since H Λ satisfies Dirichlet boundary condi-tions at both a and b , the Green’s function can be written in terms of the solutions u a = u a ( · , E,
0) and u b = u b ( · , E,
0) if E is not in the spectrum of H Λ . In this case(25) G Λ ( s, t ; E ) = 1 W ( u a , u b ) (cid:26) u a ( s ) u b ( t ) if s ≤ t,u a ( t ) u b ( s ) if s > t .where W ( u a , u b ) = u a u ′ b − u ′ a u b is the Wronskian of the solutions u a and u b . Let us first consider thecase x < y . As explained in Section 1, a fixed E is almost surely in the resolvent set of H Λ , and hence,for almost every ω , we have that k χ x G Λ ( E ) χ y k = Z x +1 x Z y +1 y (cid:12)(cid:12)(cid:12) u a ( s ) u b ( t ) W ( u a , u b ) (cid:12)(cid:12)(cid:12) dt ds (26) ≤ | W ( u a , u b ) | Z x +1 x Z y +1 y | R a ( s, E, R b ( t, E, | dt ds ≤ C | W ( u a , u b ) | | R a ( x, E, R b ( y, E, | . Here (45) in Lemma 5.1 in the Appendix was used, where a uniform constant can be chosen since W + V ω − E has local L -bounds which can be chosen uniformly in ω and E ∈ I . If x = y , thenthe representation (25) leads to two terms in (26), but Lemma 5.1 leads to the same resulting bound.Therefore, we have that E ( k χ x G Λ ( E ) χ y k s ) ≤ C E (cid:18) R sa ( x, E, R sb ( y, E, | W ( u a , u b ) | s (cid:19) (27) = C b E (cid:18)Z η max η min R sa ( x, E, R sb ( y, E, | u ′ a ( x ) u b ( x ) − u a ( x ) u ′ b ( x ) | s ρ ( η x ) dη x (cid:19) , where b E denotes the expectation with respect to the random variables { η n } n ∈ Z \{ x } .By construction, the random variable η x multiplies the single site with support on [ x, x + 1], andtherefore both R sa ( x, E,
0) and R sb ( y, E,
0) are independent of η x . From this, we conclude that E ( k χ x G Λ ( E ) χ y k s ) ≤ C b E (cid:18) R sb ( y, E, R sb ( x, E, Z η max η min ρ ( η x ) | sin( φ b ( x, E, − φ a ( x, E, | s dη x (cid:19) . The inner integral above may be bounded using Lemma 3.2 which is proven below. Using this result,we find that(28) E ( k χ x G Λ ( E ) χ y k s ) ≤ C b E (cid:18) R sb ( y, E, R sb ( x, E, (cid:19) . It follows from the definition of Pr¨ufer variables that R b ( x, E,
0) = R b ( y, E, R y ( x, E, φ b ( y, E, , and therefore, the right hand side of (28) can be written in terms of the product of transfer matrices R sb ( y, E, R sb ( x, E,
0) = 1 R sy ( x, E, φ b ( y, E, (cid:13)(cid:13)(cid:13)(cid:13) T ( x, y, E ) (cid:18) sin φ b ( y, E, φ b ( y, E, (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) − s . NOTE ON FRACTIONAL MOMENTS FOR THE ONE-DIMENSIONAL CONTINUUM ANDERSON MODEL 7 T ( x, y, E ) depends on the random variables η x , . . . , η y − , while φ b ( y, E,
0) depends on η y , η y +1 , . . . .Thus Lemma 3.1 (which holds equally well for the “backwards” transfer matrices considered here) canbe applied to the right-hand-side of (29), yielding (5) as claimed. (cid:3) This completes the proof of Theorem 1.1 given Lemma 3.2. We now state and prove this fact.
Lemma 3.2.
For any bounded interval I ⊂ R and < s < , there exists C < ∞ , such that (30) Z η max η min ρ ( η x ) | sin( φ b ( x, E, − φ a ( x, E, | s dη x ≤ C for any integer interval [ a, b ] , any integer x ∈ [ a, b ] , and E ∈ I .Proof. Observe that the random variable φ a ( x, E,
0) is determined by the parameters { η n } x − n = a , whereas φ b ( x, E,
0) depends on { η n } b − n = x . This suggests the change of variables t ( η x ) = φ b ( x, E, t ′ ( η x ) = 1 R b ( x, E, Z x +1 x f x ( t ) u b ( t, E, dt. Using the condition (3) on the single site potential in combination with Lemmas 5.1 and 5.2 we findconstants such that(31) C R b ( x, E, ≤ Z x +1 x f x ( t ) u b ( t, E, dt ≤ C R b ( x, E, < C ≤ t ′ ( η x ) ≤ C < ∞ uniformly in ω and E ∈ I . Therefore, we have that(33) Z η max η min ρ ( η x ) | sin( φ b ( x, E, − φ a ( x, E, | s dη x ≤ C k ρ k ∞ Z t ( η max ) t ( η min ) | sin( t − φ a ( x, E, | s dt. But by (32) we also have | t ( η max ) − t ( η min ) | ≤ C uniformly in ω and E ∈ I . The inequality claimedin (30) now follows using (33) and the fact that the resulting integrand has only a finite number ofintegrable singularities in any bounded interval, independent of the phase shift φ a ( x, E, (cid:3) We end this section with some comments on the proof of Theorem 1.2 , which follows by a slightadaptation of the proof of Theorem 1.1 in [1]. Essentially, this amounts to avoiding use of the coveringcondition for the single site potential required in [1] and thus allowing for single site potentials of smallsupport as in (3).To prove (6) for given E ∈ R , we may again work on the interval I = [ E , E ] with E as above. Asin Section 2 of [1] define, for a finite interval Λ and integers x , y ,(34) Y Λ ( I ; x, y ) := sup (cid:8) k χ x f ( H Λ ) χ y k : f ∈ C c ( I ) , k f k ∞ ≤ (cid:9) , where C c ( I ) are the continuous functions with compact support inside I . Let E n and ψ n denote theeigenvalues and corresponding orthonormal eigenfunctions of H Λ and P ψ n be the orthogonal projectoronto ψ n . Thus f ( H Λ ) = P n : E n ∈ I f ( E n ) P ψ n and Y Λ ( I ; x, y ) ≤ X n : E n ∈ I k χ x P ψ n χ y k (35) = X n : E n ∈ I k χ x ψ n kk χ y ψ n k As in (31), using Lemmas 5.1 and 5.2, we have k f / y ψ n k = Z y +1 y f y ( t ) ψ n ( t ) dt ≥ C ( | ψ n ( y ) | + | ψ ′ n ( y ) | ) EMAN HAMZA, ROBERT SIMS AND G¨UNTER STOLZ and k χ y ψ n k ≤ C ( | ψ n ( y ) | + | ψ ′ n ( y ) | )uniformly in Λ, n and ω . Thus k χ y ψ n k ≤ C k f / y ψ n k and (35) gives Y Λ ( I ; x, y ) ≤ CQ ( I ; x, y ), withthe eigenfunction correlator Q ( I ; x, y ) := X n : E n ∈ I k χ x ψ n kk f / y ψ n k . From here the proof is completed as in [1], where no additional use of the covering condition is made.4.
The discrete case
The one-dimensional discrete Anderson model h = h ( ω ) acts on l ( Z ) as(36) ( hu )( n ) = − u ( n + 1) − u ( n −
1) + η n ( ω ) u ( n ) . As before, we assume that the random variables ( η n ) are i.i.d. with density ρ satisfying (4). For a, b ∈ Z , a < b , we write [ a, b ] := { a, a + 1 , ..., b } , for convenience. The restriction of h to ℓ ([ a, b ]) is denoted by h [ a,b ] , the Green function by G [ a,b ] ( x, y ; z ) := h e x , ( h [ a,b ] − z ) − e y i .The following result was first proven by Minami in an appendix of [18]. We include it here tosupplement our main result Theorem 1.1 with its discrete analogue and to provide a somewhat differentself-contained proof. Theorem 4.1.
There exists a number s ∈ (0 , such that for all < s ≤ s , the bound (37) E (cid:0) | G [ a,b ] ( x, y ; E ) | s (cid:1) ≤ C e − η | x − y | , holds for all x, y ∈ [ a, b ] and E ∈ R . Here the numbers C > and η > depend on s , however, theymay be chosen independent of [ a, b ] . For E outside the spectrum of h ω exponential decay of Green’s function follows from deterministicCombes-Thomas bounds. Thus it will suffice to show (37) for energies E in, say, I = [ − η min , η max ].We start by establishing a uniform a priori bound on the left hand side of (37). This is well knownever since the ground breaking work [3], but we opt to include a somewhat streamlined proof, using amore recent change of variables idea. Lemma 4.2.
Let s ∈ (0 , . There exists a number C < ∞ such that (38) E (cid:0) | G [ a,b ] ( x, y ; E ) | s (cid:1) ≤ C, for all integers a < b and x, y ∈ [ a, b ] and E ∈ R .Proof. For x, y ∈ [ a, b ], x = y , write h = ˆ h + η x P x + η y P y , where P x = h e x , ·i e x , P y = h e y , ·i e y . Alsowriting P = P x + P y we get, using Krein’s formula,(39) G [ a,b ] ( x, y ; E ) = (cid:20) A − + (cid:18) η x η y (cid:19)(cid:21) − ( x, y ) , with the 2 × A = P (ˆ h − E ) − P .We introduce the change of variables α = ( η x + η y ), β = ( η x − η y ). With the self adjoint matrices A β := A − + β (cid:18) − (cid:19) , the right hand side of (39) becomes [ A β + αI ] − ( x, y ). Therefore, Z η max η min Z η max η min | G [ a,b ] ( x, y ; E ) | s dµ ( η x ) dµ ( η y )(40) ≤ || ρ || ∞ Z ( η max − η min ) / − ( η max − η min ) / Z η max η min k [ A β + αI ] − k s dα dβ. NOTE ON FRACTIONAL MOMENTS FOR THE ONE-DIMENSIONAL CONTINUUM ANDERSON MODEL 9
A general fact, see e.g. Lemma 4.1 of [14], says that there is a constant C = C ( s, η max , η min ) suchthat(41) Z η max η min (cid:13)(cid:13)(cid:13) [ B + α I ] − (cid:13)(cid:13)(cid:13) s dα ≤ C for all dissipative 2 × B (i.e. matrices with Im B ≥ x = y . The diagonal case x = y is easier since no change of variable is required and Krein’s formula directly reduces the claim to theelementary analogue of (41) for 1 × (cid:3) Proof. (of Theorem 4.1) Without loss of generality we assume that x < y , using the resolvent identitywe see that G [ a,b ] ( x, y ; E ) = [1 + G [ a,b ] ( x, x − E )] G [ x,b ] ( x, y ; E ) . It suffices to prove the exponential decay of E (cid:0) | G [ x,b ] ( x, y ; E ) | s (cid:1) for s ≤ s . Using Lemma 4.2 andH¨older’s inequality it then follows that (37) holds for s ≤ s / G [ x,b ] ( s, t ; E ) = 1 W ( u x , u b ) (cid:26) u x ( s ) u b ( t ) if s ≤ t,u x ( t ) u b ( s ) if s > t. Here u x and u b are the solutions of − u ( n − − u ( n + 1) + η n u ( n ) = Eu ( n ) with u x ( x −
1) = 0, u x ( x ) = 1, u b ( b ) = 1, u b ( b + 1) = 0. The constant Wronskian of u x and u b is given by W ( u x , u b )( n ) = u x ( n + 1) u b ( n ) − u x ( n ) u b ( n + 1) . Evaluating the Wronskian at n = x and denoting by b E the expectation conditioned on η x , we obtainthat E (cid:0) | G [ x,b ] ( x, y ; E ) | s (cid:1) = b E (cid:18)Z η max η min | u b ( y ) | s | u b ( x + 1) + ( E − η x ) u b ( x ) | s ρ ( η x ) dη x (cid:19) . Now the main task is to show that(43) Z η max η min | u b ( y ) | s | u b ( x + 1) + ( E − η x ) u b ( x ) | s ρ ( η x ) dη x ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) u b ( y ) u b ( y + 1) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) s (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) u b ( x ) u b ( x + 1) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) s . Expressed in terms of the discrete transfer matrices T ( x, y, E ), the right hand side is equal to C k ( u b ( y ) , u b ( y + 1)) t k s / k T ( x, y, E )( u b ( y ) , u b ( y + 1)) t k s . Thus the required bound follows from (43) andLemma 5.1 of [7], the discrete analogue of Lemma 3.1.In order to prove (43), we first note that u b ( x ), u b ( x + 1) as well as u b ( y ) are all independent of η x .With this in mind the proof of (43) is naturally divided into two cases Case I: u b ( x ) = 0, in this case, the left hand side of (43) is simply | u b ( y ) /u b ( x + 1) | s which isbounded above by k ( u b ( y ) , u b ( y + 1)) t k s / k ( u b ( x ) , u b ( x + 1)) t k s . Case II: If u b ( x ) = 0, let M = sup {| E − η | : η ∈ [ η min , η max ] , E ∈ I } . If | u b ( x + 1) /u b ( x ) | > M ,then (cid:12)(cid:12)(cid:12)(cid:12) u b ( y ) u b ( x ) (cid:12)(cid:12)(cid:12)(cid:12) s Z η max η min ρ ( η x ) | u b ( x +1) u b ( x ) + E − η x | s dη x ≤ s k ρ k ∞ ( η max − η min ) (cid:12)(cid:12)(cid:12)(cid:12) u b ( y ) u b ( x + 1) (cid:12)(cid:12)(cid:12)(cid:12) s ≤ s k ρ k ∞ ( η max − η min ) (cid:18) M (cid:19) s/ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) u b ( y ) u b ( y + 1) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) s (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) u b ( x ) u b ( x + 1) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) s . On the other hand if | u b ( x + 1) /u b ( x ) | ≤ M , using that for any β ∈ C we have(44) Z η max η min | β − η x | s ρ ( η x ) dη x ≤ C ( s, ρ ) , we see that (cid:12)(cid:12)(cid:12)(cid:12) u b ( y ) u b ( x ) (cid:12)(cid:12)(cid:12)(cid:12) s Z η max η min ρ ( η x ) | u b ( x +1) u b ( x ) + E − η x | s dη x ≤ C ( s, ρ ) (cid:0) M (cid:1) s/ | u b ( y ) | s (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) u b ( x ) u b ( x + 1) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) s ≤ C ( s, M, ρ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) u b ( y ) u b ( y + 1) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) s (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) u b ( x ) u b ( x + 1) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) s We have thus established (43), which ends the proof. (cid:3) Remarks (i) The proof of Lemma 4.2 works for multi-dimensional discrete Anderson models without anychanges.(ii) With only minor changes the proofs of Lemma 4.2 and Theorem 4.1 extend to complex energy. Inparticular, this uses that the bound (41) holds uniformly in all dissipative matrices B (as the matrices A β are now dissipative) and that (44), the scalar version of (41), holds uniformly in β ∈ C .As a consequence, we see that the exponential decay bound (37) holds uniformly in E ∈ C .Working at complex energy our arguments in Section 4 may also be used to establish the analogueof (37) for infinite volume, i.e. to show that E ( | G ( x, y ; E + iε ) | s ) ≤ Ce − η | x − y | holds uniformly in E ∈ R , ε = 0, where G ( x, y ; z ) = h e x , ( h − z ) − e y i . The only change is that u b in (42)is replaced by u ∞ , the unique solution (up to a scalar) of − u ( n − − u ( n + 1) + η n u ( n ) = ( E + iε ) u ( n )which is square-summable at + ∞ .(iii) While we expect that Theorem 1.1 extends to complex energy as well, we do not know how toget this with our method of proof. The main problem here is that the Pr¨ufer formalism strongly hingeson working with real-valued solutions. Due to its usefulness in applications, it would be interesting tofind a different argument to allow for this extension. Appendix: Basic facts
In this section, we will collect some basic facts about Pr¨ufer variables and two basic a-priori solutionestimates which we use repeatedly throughout the main text. A priori solution estimates like Lemma 5.1and Lemma 5.2 are standard tools in the theory of Sturm-Liouville operators. Lemma 5.4 as well as itsCorollary 5.5 have been frequently used in connection with spectral averaging techniques, e.g. [8]. Weprovide their proofs merely to make the paper self-contained.Throughout this appendix, with the exception of the last corollary, the energy parameter E will beabsorbed in the potential term. Lemma 5.1.
For every q ∈ L ( R ) , every interval [ c, d ] , and every solution u of − u ′′ + qu = 0 on [ c, d ] one has that ( | u ( c ) | + | u ′ ( c ) | ) exp − Z dc | q ( x ) | dx ! ≤ | u ( d ) | + | u ′ ( d ) | (45) ≤ (cid:0) | u ( c ) | + | u ′ ( c ) | (cid:1) exp Z dc | q ( x ) | dx ! . Proof.
Setting R ( t ) := | u ( t ) | + | u ′ ( t ) | , one easily calculates that R ′ ( t ) = 2Re h (1 + q ( t )) u ( t ) u ′ ( t ) i , and hence(46) | R ′ ( t ) | ≤ | q ( t ) | R ( t ) . NOTE ON FRACTIONAL MOMENTS FOR THE ONE-DIMENSIONAL CONTINUUM ANDERSON MODEL 11
Since (46) bounds the derivative of the logarithm of R ( t ), the lemma is proven. (cid:3) Lemma 5.2.
For any positive real numbers ℓ and M there exists C > such that (47) Z c + ℓc | u ( t ) | dt ≥ C (cid:0) | u ( c ) | + | u ′ ( c ) | (cid:1) for every c ∈ R , every L -function q with R c + ℓc | q ( t ) | dt ≤ M , and any solution u of − u ′′ + qu = 0 on [ c, c + ℓ ] .Proof. First, we observe that, by rescaling, it is sufficient to prove (47) for real valued solutions with | u ( c ) | + | u ′ ( c ) | = 1. By Lemma 5.1, there are constants 0 < C , C < ∞ , depending only on ℓ and M for which any real-valued solution of − u ′′ + qu = 0 satisfies C ≤ | u ( x ) | + | u ′ ( x ) | ≤ C , for all x ∈ [ c, c + ℓ ]; given the above mentioned normalization. With C := ( C / / and C := (2 C ) / ,we also have that(48) C ≤ | u ( x ) | + | u ′ ( x ) | ≤ C . We now claim that for every 0 < α < ℓ (2 + ℓ ) − exists an x ( α ) = x ∈ [ c, c + ℓ ] for which(49) | u ( x ) | ≥ α C . If, for such a fixed value of α , this is not the case, then for all x ∈ [ c, c + ℓ ], | u ( x ) | < αC , and from (48) we may also conclude that | u ′ ( x ) | ≥ C − | u ( x ) | > (1 − α ) C > . Hence the derivative, u ′ , is strictly signed. With this we may estimate,2 αC > | u ( c + ℓ ) − u ( c ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z c + ℓc u ′ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Z c + ℓc | u ′ ( x ) | dx> (1 − α ) C ℓ. This contradicts the initial assumption on the range of α , and we have proven (49).The bound (47) now follows as | u ( x ) − u ( x ) | ≤ Z xx | u ′ ( t ) | dt ≤ C | x − x | , implies that, in particular, | u ( x ) | ≥ αC / x ∈ [ c, c + ℓ ] for which | x − x | ≤ αC / (2 C ). (cid:3) Our remaining results relate to Pr¨ufer variables. In general, for any real potential q ∈ L ( R ) andreal parameters c and θ let u c be the solution of − u ′′ + qu = 0with u c ( c ) = sin θ , u ′ c ( c ) = cos θ . By regarding this solution and its derivative in polar coordinates, wedefine the Pr¨ufer amplitude R c ( x ) and the Pr¨ufer phase φ c ( x ) by writing(50) u c ( x ) = R c ( x ) sin φ c ( x ) and u ′ c ( x ) = R c ( x ) cos φ c ( x ) . For uniqueness of the Pr¨ufer phase we declare φ c ( c ) = θ and require continuity of φ c in x . In whatfollows the initial phase θ will be fixed and we thus leave the dependence of u c , R c and φ c implicit inour notation.In the new variables R and φ the second order equation − u ′′ + qu = 0 becomes a system of two firstorder equations, where the equation for φ is not coupled with R : Lemma 5.3.
For fixed c and θ , one has that (51) (ln R c ( x )) ′ = (1 + q ( x )) sin (2 φ c ( x )) , and (52) φ ′ c ( x ) = 1 − (1 + q ( x )) sin ( φ c ( x )) . Proof.
It is clear that R c = u + ( u ′ ) , and (51) follows from a simple calculation. To see (52), observethe following two equations: u ′ = R ′ c sin( φ c ) + R c cos( φ c ) φ ′ c and qu = u ′′ = R ′ c cos( φ c ) − R c sin( φ c ) φ ′ c .Solving for φ ′ c yields (52). (cid:3) We have the following formula for the derivative of the Pr¨ufer phase with respect to a couplingconstant at a potential.
Lemma 5.4.
Let W and V be real valued functions in L ( R ) . For real parameters c , θ and λ , let u c be the solution of − u ′′ + W u + λV u = 0 normalized so that u c ( c ) = sin( θ ) and u ′ c ( c ) = cos( θ ) . Denoting the Pr¨ufer variables of u c by φ c ( x, λ ) and R c ( x, λ ) , indicating their dependence on the coupling constant λ , one has that (53) ∂∂λ φ c ( x, λ ) = − R − c ( x, λ ) Z xc V ( t ) u c ( t, λ ) dt. Proof.
Using both (51) and (52) from Lemma 5.3 above, one finds that ∂ ∂λ∂x φ c ( x, λ ) = − V ( x ) sin ( φ c ( x, λ )) − ∂∂x ln (cid:2) R c ( x, λ ) (cid:3) ∂∂λ φ c ( x, λ ) , This implies that(54) ∂∂x (cid:18) R c ( x, λ ) ∂∂λ φ c ( x, λ ) (cid:19) = − V ( x ) R c ( x, λ ) sin ( φ c ( x, λ )) = = − V ( x ) u c ( x, λ ) , for almost every pair ( x, λ ). Since ∂∂λ φ c ( c, λ ) = 0, (53) follows immediately from (54). (cid:3) As a special case one finds the energy derivative of the Pr¨ufer phase.
Corollary 5.5.
Let u be the solution of − u ′′ + W u = Eu normalized so that u ( c ) = sin( θ ) and u ′ ( c ) = cos( θ ) , and let φ c ( x, E ) and R c ( x, E ) be the corresponding Pr¨ufer variables. Then (55) ∂∂E φ c ( x, E ) = R − c ( x, E ) Z xc u ( t ) dt. Proof.
This follows from Lemma 5.4 by setting V constant to − (cid:3) Acknowledgements:
A part of this work was supported by the National Science Foundation, e.g.,R.S. under Grant
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