Holder continuity of absolutely continuous spectral measures for one-frequency Schrodinger operators
aa r X i v : . [ m a t h . D S ] D ec H ¨OLDER CONTINUITY OF ABSOLUTELY CONTINUOUS SPECTRALMEASURES FOR ONE-FREQUENCY SCHR ¨ODINGER OPERATORS
ARTUR AVILA AND SVETLANA JITOMIRSKAYA
Abstract.
We establish sharp results on the modulus of continuity of the distribution of thespectral measure for one-frequency Schr¨odinger operators with Diophantine frequencies in theregion of absolutely continuous spectrum. More precisely, we establish 1 / Introduction
In this work we study absolutely continuous spectral measures of (one-frequency) quasiperiodicSchr¨odinger operators H = H λv,α,θ defined on ℓ ( Z )(1.1) ( Hu ) n = u n +1 + u n − + λv ( θ + nα ) u n where v is the potential, λ ∈ R is the coupling constant, α ∈ R \ Q is the frequency and θ ∈ R is thephase. A central example is given by the almost Mathieu operator, when v ( x ) = 2 cos(2 πx ).Except where otherwise noted, below we assume the frequency α to be Diophantine in the usualsense (see definition in Section 3), and v analytic.Absolutely continuous spectrum occurs only rarely in one-dimensional Schr¨odinger operators [R].Until recently it was expected that in the class of ergodic Schr¨odinger operators it only occursfor almost periodic potentials, a conjecture recently disproved [A2]. Quasiperiodic operators withanalytic potential stand out in this respect as the family { H λv,α,θ } λ ∈ R , for small couplings λ, isalways in the metallic phase (has good transport properties) with zero Lyapunov exponents andabsolutely continuous spectrum.We will be concerned with the regularity of spectral measures. More precisely, given a function f ∈ ℓ ( Z ) with k f k = 1, and letting µ f = µ fλv,α,θ be the associated spectral measure , what can besaid of the modulus of continuity of the distribution of µ f ? We will assume that f is a reasonablylocalized function in the sense that f ∈ ℓ ( Z ) (notice that without regularity assumptions, thereare no non-trivial restrictions on µ f : any probability measure absolutely continuous with respect tosome spectral measure is still a spectral measure). Our first result concerns small potentials: Theorem 1.1.
For every v ∈ C ω ( R / Z , R ) , there exists λ = λ ( v ) > such that if | λ | < λ and α is Diophantine then µ fλv,α,θ ( J ) ≤ C ( α, λv ) | J | / k f k ℓ , for all intervals J and all θ. For the almostMathieu operator, one can take λ = 1 .Remark . The smallness constant λ only depends on bounds on the analytic extension of v tosome band |ℑ x | < ǫ . This is important for applications to arbitrary potentials (see below). The † This work was supported in part by NSF, grant DMS-0601081, and BSF, grant 2006483. This research waspartially conducted during the period A.A. served as a Clay Research Fellow. That is, µ f ( X ) = k Π X ( f ) k where Π X : ℓ ( Z ) → ℓ ( Z ) is the spectral projection associated to the Borel set X ⊂ R . constant C depends on bounds on the analytic extension of λv and on the Diophantine propertiesof α .Recall that averaging the distributions of spectral measures with respect to the phase θ yieldsthe integrated density of states (i.d.s.), whose regularity is therefore significantly simpler to analyze.Indeed in [AJ], it is shown that the i.d.s. is 1 / / λ >
1, [J], [AJ]), and hence discontinuous distributions. The keypoint of Theorem 1.1 is that here we are able to control the behavior of each individual spectralmeasure, uniformly on θ .The study of small potentials is not merely interesting on its own: it gives information aboutthe absolutely continuous spectrum of an arbitrary potential. To make this precise, one introducesthe notion of almost reducibility : roughly speaking an energy is almost reducible if the associatedcocycle ( α, A ( E − λv ) ) (a dynamical system(1.2) ( x, w ) ( x + α, A ( E − λv ) · w ) , (1.3) A ( E − λv ) = (cid:18) E − λv ( x ) −
11 0 (cid:19) , that describes the behavior of solutions of the eigenvalue equation H λv,α,θ u = Eu ), is analyticallyconjugate (in a uniform band) to the associated cocycle of some ( α, A ( E ′ − v ′ ) ) with v ′ arbitrarilysmall. It follows from renormalization [AK1], [AK2], that almost reducible energies (indeed reducible energies, for which v ′ can be taken as 0) form an essential support of absolutely continuous spectrum.In [AJ], almost reducibility was proved for all energies in the case of small potentials (indeed thesame setting of Theorem 1.1), which implies that almost reducibility is stable (in particular, the setof almost reducible energies is open). Theorem 1.2.
Let v ∈ C ω ( R / Z , R ) and let α be Diophantine. Then for any almost reducible energy E ∈ Σ v,α , (thus for a.e. energy in Σ acv,α ) there exists C, ǫ > such that if J ⊂ ( E − ǫ, E + ǫ ) is aninterval then, for all θ, µ fv,α,θ ( J ) ≤ C | J | / k f k ℓ . As far as we know Theorems 1.2,1.1 are the first results on fine properties of individual absolutelycontinuous spectral measures of ergodic operators.Let us call attention to the following conjecture that clarifies the fundamental importance ofunderstanding almost reducibility:
Spectral Dichotomy Conjecture.
For typical v, α, θ , H v,α,θ is the direct sum of operators H + and H − with disjoint spectra such that H + is “localized” and H − is “almost reducible”. Remark . (1) Typical should be understood in the measure-theoretical sense of prevalence .In particular frequencies may be assumed to be Diophantine.(2) Localization for H + means both what is usually understood as Anderson localization (purepoint spectrum with exponentially decaying eigenfunctions) or dynamical localization. Al-Explain most reducibility for H − just means that the spectrum of H − is the closure of almostreducible energies for H , but as described above, it indeed provides a very fine spectral de-scription: in particular the results of [AJ] and this paper apply to H − , e.g., it has absolutelycontinuous spectral measures with 1 / H + / H − must be defined by spec-tral projection on the parts of the spectrum where the associated cocycle has positive/zeroLyapunov exponent(1.4) L ( E ) = lim 1 n Z R / Z ln k A ( E − v ) n ( x ) k dx, ONTINUITY OF SPECTRAL MEASURES 3 (1.5) A ( E − v ) n ( x ) = A ( E − v ) ( x + ( n − α ) · · · A ( E − v ) ( x ) . The result of disjointness of the spectra for this decomposition was recently established (inthe typical setting) [A3], [A4].(4) With H + defined as above, the precise spectral and dynamical description, particularly dy-namical localization, follows (in the typical setting) from a minoration of the Lyapunovexponent through the spectrum of H + , using [BG] and [BJ3]. Such minoration is a conse-quence of disjointness of spectra [A3, A4] and continuity of the Lyapunov exponent [BJ1].More is known in this regime [JL3],[GS2], [GS3].(5) What is still incomplete in the above picture is the description of H − . Zero Lyapunovexponent does not necessarily imply almost reducibility (consider the critical almost Mathieuoperator). In [A3], [A4], it is shown that (in the typical setting) energies in the spectrum ofof H − satisfy not only L ( E ) = 0 but the stronger condition (called subcriticality )(1.6) ln k A n ( x ) k = o ( n )uniformly in some band |ℑ x | < ǫ . The Spectral Dichotomy Conjecture is thus reduced to theAlmost Reducibility Conjecture (the main outstanding problem in the theory): subcriticalityimplies almost reducibility.1.1. Further perspective.
One should distinguish between two possible regimes of small | λ | (sim-ilar considerations can be applied to the analysis of large coupling). One is perturbative , meaningthat the smallness condition on | λ | depends not only on the potential v , but also on the frequency α :the key resulting limitation is that the analysis at a given coupling, however small, has to excludea positive Lebesgue measure set of α . Such exclusions are inherent to the KAM-type methods thathave been traditionally used in this context. The other, stronger regime, is called non-perturbative ,meaning that the smallness condition on | λ | only depends on the potential, leading to results thathold for almost every α. A thorough study of absolutely continuous spectrum of operators (1.1) in the case of small an-alytic potentials in the perturbative regime was done by Eliasson [E]. He proved the reducibilityof the associated cocycle for almost all energies in the spectrum and fine estimates on solutions forthe other energies, by developing a sophisticated KAM scheme, which avoided the limitations ofearlier KAM methods (that go back to the work of Dinaburg-Sinai [DiS] and that excluded partsof the spectrum from consideration). This allowed him in particular to conclude purely absolutelycontinuous spectrum.A thorough study of absolutely continuous spectrum of operators (1.1) in the non-perturbativeregime of [BJ2] was done in [AJ] where we used some techniques of [BJ2] to obtain localizationestimates for all energies for the dual model, and developed quantitative Aubry duality theory,which allowed us, in particular, to conclude almost reducibility for all energies (including thosefor which neither dual localization nor reducibility hold). The smallness condition on the couplingconstant in [AJ] coincides with that of [BJ2]. In particular, for the almost Mathieu operator, all theestimates and conclusions hold throughout the subcritical regime λ < . The analyses of [E] and [AJ] allowed to obtain sharp bounds (H¨older-1/2 continuity) for theintegrated density of states, for Diophantine frequencies. This was done, in perturbative and non-perturbative regimes in correspondingly [Am] and [AJ].Earlier, Goldstein-Schlag [GS2] had shown H¨older continuity of the integrated density of statesfor a full Lebesgue measure subset of Diophantine frequencies in the regime of positive Lyapunovexponents, with the result becoming almost sharp for the super-critical almost Mathieu operator:(1 / − ǫ )-H¨older for any ǫ, and | λ | > | λ | < , by duality. Before that Bourgain [B1] had obtained almost 1 / ARTUR AVILA AND SVETLANA JITOMIRSKAYA continuity for almost Mathieu type potentials in the perturbative regime, for Diophantine α andln | λ | large (depending on α ).There were no results however, neither recently nor previously, on the modulus of continuity ofthe individual spectral measures, even in the perturbative regime. In this paper we achieve thisby applying methods developed in [AJ] combined with a dynamical reformulation of the power-lawsubordinacy techniques of [JL2],[JL3]. As mentioned above, our all energy results hold throughoutthe regime of [BJ2], and in particular, for all sub-critical almost-Mathieu operators. The generalabsolutely continuous case is obtained through a reduction to the small potential case and almostreducibility result of [AK1].Our estimate is optimal in several ways. First, there are square-root singularities at the boundariesof gaps (e.g., [P2]), so the modulus of continuity cannot be improved. Also, since the integrateddensity of states satisfies(1.7) N ( E ) = lim n →∞ n n − X k =0 µ σ k ( f ) (0 , E ]( σ : ℓ ( Z ) → ℓ ( Z ) denotes the shift), the spectral measures of ℓ functions cannot have higher mod-ulus of continuity than N ( E ). There are examples with lower regularity of N ( E ) that demonstratethat Diophantine condition on α as well as a condition on λ are essential here. In particular, it isknown that for the almost Mathieu operator for a certain non-empty set of α which satisfy goodDiophantine properties (but has zero Lebesgue measure) and λ = 1, the integrated density of statesis not H¨older ([B3], Remark after Corollary 8.6). Additionally, for any λ = 0 and generic α , theintegrated density of states is not H¨older (this is because the Lyapunov exponent is discontinuousat rational α , which easily implies that it is not H¨older for generic α . Such discontinuity holds forthe almost Mathieu operator and presumably generically). Remark . As our approach is non-perturbative and non-KAM, it is not expected to break down atthe Brjuno condition and can potentially be extended much further. While, as mentioned above, theexact modulus of continuity should depend on the Diophantine properties for very well approximated α we expect the same methods to work for small rate of exponential approximation as well. We donot pursue it here though. 2. Preliminaries
For a bounded analytic (perhaps matrix valued) function f defined on a strip {|ℑ z | < ǫ } andextending continuously to the boundary, we let k f k ǫ = sup |ℑ z | <ǫ | f ( z ) | . If f is a bounded continuousfunction on R , we let k f k = sup x ∈ R | f ( x ) | .2.1. Cocycles.
Let α ∈ R \ Q , A ∈ C ( R / Z , SL(2 , C )). We call ( α, A ) a (complex) cocycle . The Lyapunov exponent is given by the formula(2.1) L ( α, A ) = lim n →∞ n Z ln k A n ( x ) k dx, where A n , n ∈ Z , is defined by ( α, A ) n = ( nα, A n ), so that for n ≥ A n ( x ) = A ( x + ( n − α ) · · · A ( x ) . We say that ( α, A ) is uniformly hyperbolic if there exists a continuous splitting C = E s ( x ) ⊕ E u ( x ), x ∈ R / Z such that for some C > c >
0, and for every n ≥ k A n ( x ) · w k ≤ Ce − cn k w k , w ∈ E s ( x )and k A − n ( x ) · w k ≤ Ce − cn k w k , w ∈ E u ( x ). In this case, of course L ( α, A ) > ONTINUITY OF SPECTRAL MEASURES 5
Given two cocycles ( α, A (1) ) and ( α, A (2) ), a (complex) conjugacy between them is a continuous B : R / Z → SL(2 , C ) such that(2.3) A (2) ( x ) = B ( x + α ) A (1) ( x ) B ( x ) − . We assume now that ( α, A ) is a real cocycle, that is, A ∈ C ( R / Z , SL(2 , R )). The notionof real conjugacy (between real cocycles) is the same as before, except that we ask for B ∈ C ( R / Z , PSL(2 , R )). Real conjugacies still preserve the Lyapunov exponent.We say that a real cocycle ( α, A ) is (analytically) reducible if it is (real) conjugate to a constantcocycle, and the conjugacy is analytic. We say that it is almost reducible if there exists a sequence A ( n ) ∈ C ω ( R / Z , R ) converging (uniformly in some band {|ℑ z | < ǫ } ) to a constant, such that( α, A ( n ) ) is conjugated to ( α, A ), and the conjugacies extend holomorphically to some fixed band: B ( n ) ∈ C ωǫ ( R / Z , PSL(2 , R )). Schr¨odinger operators.
We consider now Schr¨odinger operators { H v,α,θ } θ ∈ R (we incorporatethe coupling constant into v ). The spectrum Σ = Σ v,α does not depend on θ , and it is the set of E such that ( α, A ( E − v ) ) is not uniformly hyperbolic, with A ( E − v ) as in the introduction.For f ∈ l ( Z ) the spectral measure µ = µ fx is defined so that(2.4) h ( H x − E ) − f, f i = Z R E ′ − E dµ ( E ′ )holds for E in the resolvent set C \ Σ. Alternatively, for a Borel set X ,(2.5) µ fv,α,θ ( X ) = k Π X f k , where Π X is the corresponding spectral projection.The integrated density of states is the function N : R → [0 ,
1] that can be defined by (1.7) Itis a continuous non-decreasing surjective (for bounded potentials) function. The Thouless formularelates the Lyapunov exponent to the integrated density of states(2.6) L ( E ) = Z R ln | E ′ − E | dN ( E ′ ) . Almost reducibility and the support of absolutely continuous spectrum.
We justifythe claim made in the introduction that almost reducible energies support the absolutely continuouspart of the spectral measures.By [LS], the set Σ = { L ( E ) = 0 } (which is closed by [BJ1]) is the essential support of acspectrum, so that the ac spectral measures are precisely those probability measures on Σ which areequivalent to Lebesgue. Theorem 2.1 ([AFK]) . If α ∈ R \ Q then for almost every E ∈ Σ , ( α, A ( E − v ) ) is real analyticallyconjugated to a cocycle of rotations, i.e., taking values in SO(2 , R ) . This result was proved in [AK1] under a full measure condition on α (which is stronger thanDiophantine). For Diophantine α , it can also be obtained as a consequence of [AJ] and [AK2].It is easy to see that for any α ∈ R \ Q , analytic cocycles of rotations are almost reducible.Moreover, if α is Diophantine (more generally, if the best rational approximations to α are subex-ponential), then analytic cocycles of rotations are reducible. In fact this last property is automatic, but this is non-trivial (it follows from the openness of almost reducibility).
ARTUR AVILA AND SVETLANA JITOMIRSKAYA
Almost reducibility in Schr¨odinger form.
While almost reducibility allows one to conju-gate the dynamics of the cocycle close to a constant, it is rather convenient to have the conjugatedcocycle in Schr¨odinger form, since many results (particularly the ones depending on Aubry duality,as the ones obtained in [AJ]) are obtained only in this setting. The following result takes care ofthis.
Lemma 2.2.
Let ( α, A ) ∈ R \ Q × C ω ( R / Z , SL(2 , R )) be almost reducible. Then there exists ǫ > such that for every γ > , there exists v ∈ C ωǫ ( R / Z , R ) with k v k ǫ < γ , E ∈ R and B ∈ C ωǫ ( R / Z , PSL(2 , R )) such that B ( x + α ) A ( x ) B ( x ) − = A ( E − v ) ( x ) . Moreover, for every < ǫ ≤ ǫ ,there exists δ > such that if ˜ A ∈ C ω ( R / Z , SL(2 , R )) is such that k ˜ A − A k ǫ < δ then there exists ˜ v ∈ C ω ( R / Z , R ) , such that k ˜ v k ǫ < γ and ˜ B ∈ C ωǫ ( R / Z , PSL(2 , R )) such that k ˜ B − B k < γ and ˜ B ( x + α ) ˜ A ( x ) ˜ B ( x ) − = A ( E − ˜ v ) ( x ) . For the proof, one basically just needs to be able to convert non-Schr¨odinger perturbations ofSchr¨odinger cocycles to Schr¨odinger form. This problem is studied in [A4]. For completeness, we willgive a much simpler (unpublished) argument of Avila-Krikorian which is enough for our purposes.
Lemma 2.3.
Let v ∈ C ωǫ ( R / Z , R ) and α ∈ R \ Q , be such that /v ∈ C ωǫ ( R / Z , R ) . If A ∈ C ωǫ ( R / Z , SL(2 , R )) and k A − A ( v ) k ǫ is sufficiently small (depending on k v k ǫ and k /v k ǫ ), then thereexists v ′ ∈ C ωǫ ( R / Z , R ) and B ∈ C ωǫ ( R / Z , SL(2 , R )) such that k v ′ − v k ǫ and k B − id k ǫ are small and B ( x + α ) A ( x ) B ( x ) − = A ( v ′ ) ( x ) .Proof. Let w = (cid:18) w w w − w (cid:19) ∈ C ωǫ ( R / Z , sl(2 , R )) be such that k w k ǫ is small and A = A ( v ) e w . Let s = (cid:18) s s s − s (cid:19) ∈ C ωǫ ( R / Z , sl(2 , R )) be defined by s = 0, s ( x ) = w ( x ) + w ( x ) v ( x ) , s ( x ) = − w ( x − α ) v ( x − α ) and let ˜ v ∈ C ωǫ ( R / Z , R ) be given by˜ v ( x ) = v ( x ) − w ( x ) + w ( x + α ) + w ( x + α ) v ( x + α ) + v ( x ) w ( x ) − w ( x − α ) v ( x − α ) . Then k ˜ v − v k ǫ ≤ C k w k ǫ and e s ( x + α ) A ( x ) e − s ( x ) is of the form A (˜ v ) e ˜ w where k ˜ w k ǫ ≤ C k w k ǫ , for someconstant C depending on k v k ǫ and k /v k ǫ . The result follows by iteration. (cid:3) Remark . (1) This result with only assuming v ∈ C ωǫ ( R / Z , R ) to be non-identically zero isproved in [A4].(2) For Diophantine α the result holds with no conditions on v ∈ C ωǫ ( R / Z , R ) . Proof of Lemma 2.2.
Since ( α, A ) is almost reducible, if ǫ > B ( n ) ∈ C ωǫ ( R / Z , PSL(2 , R )) and A ∗ ∈ SL(2 , R ) such that k B ( n ) ( x + α ) A ( x ) B ( n ) ( x ) − − A ∗ k ǫ →
0. Let us show that, up to changing B ( n ) to C ( n ) B ( n ) for an appropriate choice of C ( n ) ∈ C ωǫ ( R / Z , SL(2 , R )), we may assume that A ∗ is of the form (cid:18) E −
11 0 (cid:19) with E = 0. Indeed:(1) If | tr A ∗ | >
2, by converting first to the diagonal form, we can find ˜ C A ∗ ∈ SL(2 , R ) such that˜ C A ∗ A ∗ ˜ C − A ∗ = (cid:18) tr A ∗ −
11 0 (cid:19) , so we can just take C ( n ) = C A ∗ .(2) If | tr A ∗ | <
2, there exists ˜ C A ∗ ∈ SL(2 , R ) such that ˜ C A ∗ A ∗ ˜ C − A ∗ = R θ for some θ = k/ , k ∈ Z . If 0 < sin 2 πθ <
1, then let C − θ = 1(sin 2 πθ ) / (cid:18) − sin 2 πθ − cos 2 πθ (cid:19) , ONTINUITY OF SPECTRAL MEASURES 7 so that C θ R θ C − θ = (cid:18) πθ −
11 0 (cid:19) , and we can take C ( n ) = C θ ˜ C A ∗ . Otherwise, let k ∈ Z be such that 0 < sin 2 π ( θ + kα ) <
1, and take C ( n ) ( x ) = C θ + kα R kx ˜ C A ∗ .(3) If | tr A ∗ | = 2, there exists ˜ C ( n ) ∈ SL(2 , R ) such that ˜ C ( n ) A ∗ ( ˜ C ( n ) ) − → R θ , where θ = 0 or θ = 1 / . Indeed, either A ∗ is equal to such R θ or we can assume it is in the Jordan form, inwhich case one can take ˜ C ( n ) = (cid:18) ǫ n ǫ n ǫ n (cid:19) , By choosing ǫ n appropriately, we may also assume that k ˜ C ( n ) k k B ( n ) ( x + α ) A ( x ) B ( n ) ( x ) − − A ∗ k ǫ →
0. Choosing again k ∈ Z such that 0 < sin 2 π ( θ + kα ) < C ( n ) = C θ + kα R kx ˜ C ( n ) .Now the first statement follows from Lemma 2.3. For the second statement, apply again Lemma2.3. (cid:3) Estimates on the dynamics
Here we describe the [AJ] estimates on the dynamics of almost reducible cocycles.3.1.
Rational approximations.
Let q n be the denominators of the approximants of α . We recallthe basic properties:(3.1) k q n α k R / Z = inf ≤ k ≤ q n +1 − k kα k R / Z , (3.2) 1 ≥ q n +1 k q n α k R / Z ≥ / . We say that α is Diophantine if ln q n +1 ln q n = O (1). Let DC ⊂ R be the set of Diophantine numbers.3.2. Resonances.
Let α ∈ R , θ ∈ R , ǫ >
0. We say that k is an ǫ -resonance if k θ − kα k R / Z ≤ e −| k | ǫ and k θ − kα k R / Z = min | j |≤| k | k θ − jα k R / Z . Remark . In particular, there always exists at least one resonance, 0. If α ∈ DC( κ, τ ), k θ − kα k R / Z ≤ e −| k | ǫ implies k θ − kα k R / Z = min | j |≤| k | k θ − jα k R / Z for k > C ( κ, τ ).For fixed α and θ , we order the ǫ -resonances 0 = n < | n | ≤ | n | ≤ ... . We say that θ is ǫ -resonant if the set of resonances is infinite. If θ is non-resonant, with the set of resonances { n , . . . , n j } we formally set n j +1 = ∞ . The Diophantine condition immediately implies exponentialrepulsion of resonances:
Lemma 3.1. If α ∈ DC , then | n j +1 | ≥ c k θ − n j α k − c R / Z ≥ ce cǫ | n j | , where c = c ( α, ǫ ) > . ARTUR AVILA AND SVETLANA JITOMIRSKAYA
Dynamical estimates.
Let us say that a cocycle ( α, A ) ∈ R \ Q × C ω ( R / Z , SL(2 , R )) is( C, c, ǫ )-good if there exists θ ∈ R with the following property: for any finite ǫ -resonance n j associated to α and θ , denoting n = | n j | + 1 and N = | n j +1 | , there exists Φ : R / Z → SL(2 , C )analytic with k Φ k cn − C ≤ Cn C such that(3.3) Φ( x + α ) A ( x )Φ( x ) − = (cid:18) e πiθ e − πiθ (cid:19) + (cid:18) q ( x ) q ( x ) q ( x ) q ( x ) (cid:19) , with(3.4) k q k cn − C , k q k cn − C , k q k cn − C ≤ Ce − cN and(3.5) k q k cn − C ≤ Ce − cn (ln(1+ n )) − C . The following is one of the main estimates of [AJ] (combining Theorems 3.3, 3.4 and 5.1 of [AJ]):
Theorem 3.2. [see Theorems 3.4 and 5.1 of [AJ] ]There exists a constant c > with the following property. Let v ∈ C ω ( R / Z , R ) and E ∈ Σ v,α , ( α, A ) . If for some < ǫ < , k v k ǫ < c ǫ then ( α, A ) is ( C, c, ǫ ) -good for some constants c = c ( ǫ, α ) > , C = C ( ǫ, α ) > and ǫ = ǫ ( ǫ ) . A more precise result is available for the almost Mathieu operator (still a combination of Theorems3.4 and 5.1 of [AJ]):
Theorem 3.3. [see Theorems 3.4 and 5.1 of [AJ] ]For every < λ < and α Diophantine, there exists C = C ( λ , α ) , c = c ( λ , α ) , ǫ = ǫ ( λ ) > such that for v = 2 λ cos 2 π ( x + θ ) with | λ | < λ , and E ∈ Σ v,α , ( α, A ( E − v ) ) is ( C, c, ǫ ) -good. Coupling Theorem 3.2 and Lemma 2.2 we immediately get:
Theorem 3.4.
Let α be Diophantine and let A ∈ C ω ( R / Z , SL(2 , R )) . If ( α, A ) is almost reduciblethen there exists ¯ ǫ > such that for every < ǫ < ¯ ǫ there exist δ, C, c, ǫ > such that if ˜ A ∈ C ω ( R / Z , SL(2 , R )) is such that k ˜ A − A k ǫ < δ and ( α, ˜ A ) is not uniformly hyperbolic then ( α, ˜ A ) is ( C, c, ǫ ) -good.Remark . Using [A1], Theorem 3.8, one can consider a stronger definition of goodness, so thatTheorem 3.2, and hence Theorem 3.4, and Theorem 3.3, still hold: k Φ k c ≤ Cn C , k q j k c ≤ Ce − cN , j = 1 , ,
4, and k q k c ≤ Ce − cn .An immediate consequence of ( C, c, ǫ )-goodness is (see [AJ] for the easy argument): Lemma 3.5. If ( α, A ) is ( C, c, ǫ ) -good then for every s ≥ we have k A s k ≤ C ′ ( C, c, ǫ , α )(1 + s ) . Regularity of the spectral measures at good energies
Let v ∈ C ω ( R / Z , R ), E ∈ Σ v,α . Let µ x = µ e − v,α,x + µ e v,α,x and e i is the Dirac mass at i ∈ Z .Our main estimate is: Theorem 4.1. If ( α, A ( E − v ) ) is ( C , c , ǫ ) -good then for every < ǫ < , µ x ( E − ǫ, E + ǫ ) ≤ C ′ ( C , c , ǫ , α ) ǫ / .Proof of Theorems 1.1 and 1.2. We first prove Theorem 1.2. By Theorem 3.4, ( α, A ( E ′ − v ) ) is( C , c , ǫ )-good for any E ′ near E which is in the spectrum. By Theorem 4.1, we get(4.1) µ x ( J ) ≤ C ′ | J | / for any interval containing such an E ′ , and hence (since µ x is supported on the spectrum), forany interval contained in a neighborhood of E . Let σ : l ( Z ) → l ( Z ) be the shift f ( i + 1) = ONTINUITY OF SPECTRAL MEASURES 9 σf ( i ). Then σH v,α,x σ − = H v,α,x + α . Thus µ σfx + α = µ fx and µ e k x = µ e x + kα ≤ µ x + kα . By (2.5), µ fx ( E − ǫ, E + ǫ ) / defines a semi-norm on l ( Z ). Therefore, by the triangle inequality, µ fx ( J ) / ≤ P k ∈ Z | f ( k ) | ( µ x + kα ( J )) / , and the result follows immediately from (4.1).Theorem 1.1 is proved analogously, using Theorems 3.2 and 3.3 to establish appropriate ( C, c )-goodness. (cid:3)
It therefore remains to prove Theorem 4.1 which we do in Section 4.2.Through the end of this section, A = A ( E − v ) . We will use C and c for large and small constantsthat only depend on C , c , ǫ , and α .4.1. Spectral measures and m -functions. In the study of µ = µ x , we will use a result of [JL2](or its improvement in [KKL]), interpreted in terms of cocycles. In the definition of the m -functionsbelow, we follow the notation of [JL3].We will consider energies E + iǫ , E ∈ R , ǫ >
0. Then there are non-zero solutions u ± of Hu ± = ( E + iǫ ) u ± which are l at ±∞ , well defined up to normalization. We define(4.2) m ± = ∓ u ± u ± . It coincides with the Weyl-Titchmarsh m -function which is the Borel transform of the spectralmeasure µ ± = µ ± e of the corresponding half-line problem with Dirichlet boundary conditions: m ± ( z ) = Z dµ ± ( x ) x − z , (e.g.[CL]).Thus m ± has positive imaginary part for every ǫ > M ( E + iǫ ) = Z E ′ − ( E + iǫ ) dµ ( E ′ ) . Notice that M ( E + iǫ ) ∈ H = { z, ℑ z > } . We have(4.4) ℑ M ( E + iǫ ) ≥ ǫ µ ( E − ǫ, E + ǫ ) . Then, as discussed in [JL3],(4.5) M = m + m − − m + + m − . As in [JL3], we define m + β = R − β/ π · m + , or, more generally, z β = R − β/ π z. Those are Borel transforms of the half-line spectral measures µ β = µ βe of operator H on l ([0 , ∞ ))with boundary conditions u cos β + u sin β = 0 . Here we make use of the action of SL(2 , C ) on C , (cid:18) a bc d (cid:19) · z = az + bcz + d . Let ψ ( z ) = sup β | z β | . We have(4.6) ψ ( z ) − ≤ ℑ z ≤ | z | ≤ ψ ( z ) . where the first inequality easily follows from the invariance of φ , see below. It was shown in [DKL]that, as a corollary of the maximal modulus principle, one obtains(4.7) | M | ≤ ψ ( m + ) . This also can be shown directly by the following computation, that gives some more quantitativeestimates. Let φ ( z ) = 1 + | z | ℑ z . If z ∈ H then φ ( z ) ≥ φ ( z ) is invariant with respect to the action of R β . Thus the maximumof | z β | is attained when z β is purely imaginary with ℑ z β > φ ( z ) + ( φ ( z ) − / . Thus ψ ( z ) = φ ( z ) + ( φ ( z ) − / . We can compute(4.8) φ ( M ) = φ ( m + ) φ ( m − ) + 1 φ ( m + ) + φ ( m − )which implies φ ( M ) ≤ φ ( m + ) and hence(4.9) ψ ( M ) ≤ ψ ( m + )(whatever the value of m − ∈ H ). By (4.6), this gives (4.7).For k ≥ P ( k ) = k X j =1 A ∗ j − ( x + α ) A j − ( x + α ) . Then P ( k ) is an increasing family of positive self-adjoint linear maps. In particular, k P ( k ) k , det P ( k ) k P ( k ) k and det P ( k ) are increasing positive functions. It is not difficult to see that k P ( k ) k (and hence det P ( k ) )is also unbounded (since A j ∈ SL(2 , R ) implies tr P ( k ) ≥ k ). Lemma 4.2.
Let ǫ be such that det P ( k ) = ǫ . Then (4.11) C − < ψ ( m + ( E + iǫ ))2 ǫ k P ( k ) k < C. Proof.
Let ( u βj ) j ≥ satisfy(4.12) A ( x + jα ) · u βj u βj − ! = u βj +1 u βj ! ,u β cos β + u β sin β = 0 , | u β | + | u β | = 1 . For integer L define(4.13) k u k L = L X j =1 | u j | / . Unlike [JL2] it will be sufficient for us here to deal with the “discrete” definition (4.13) of k u k L , because we only deal with bounded potentials, see proof of Corollary 4.7.Theorem 1.1 of [JL2] can be stated as follows (see (2.13) in [JL3]). If k u β k L k u β + π/ k L = 12 ǫ then(4.14) 5 − √ < | m + β ( E + iǫ ) | k u β k L k u β + π/ k L < √ . ONTINUITY OF SPECTRAL MEASURES 11
In other words,(4.15) 5 − √ < | m + β ( E + iǫ ) | ǫ k u β + π/ k L < √ . It is immediate to see that if L = 2 k then(4.16) k u β k L = h P ( k ) (cid:18) u β u β (cid:19) , (cid:18) u β u β (cid:19) i ≤ k P ( k ) k , with equality for β maximizing k u β k L . Thus(4.17) det P ( k ) = inf β k u β k L k u β + π/ k L , the infimum being attained at the critical points of β
7→ k u β k L . We conclude that if k det P ( k ) k = ǫ ,then for every β , | m + β ( E + iǫ ) | ǫ k P ( k ) k < √ , and if β is such that k u β + π/ k L is maximal then | m + β ( E + iǫ ) | ǫ k P ( k ) k > − √ . This together gives (4.11) with C = 5 + √ (cid:3) Remark . Replacing (4.14) with a result of [KKL] we can obtain by the same argument that if ǫ is such that det P ( k ) = ǫ , then(4.18) 2 − √ < ψ ( m + ( E + iǫ )) ǫ k P ( k ) k < √ . We note that det P ( k ) is precisely the Hilbert-Schmidt norm of operator K in [KKL] at scale L. Theexact value of C in (4.11) is not important for the present argument.4.2. Proof of Theorem 4.1.
We need to estimate k P ( k ) k and det P ( k ) = k P ( k ) kk P − k ) k − . Moreprecisely, we will show that k P ( k ) k ≤ C k P − k ) k − , which is just enough for our purposes. Remark . By Lemma 3.5, we have k P ( k ) k ≤ Ck , so to estimate k P ( k ) k ≤ C k P − k ) k − it would beenough to show that k P − k k ≤ Ck − . We do not know whether the latter estimate holds.A key point will be to compare the dynamics of ( α, A ) with the dynamics of ( α, T ) where T is insome particularly simple triangular form. Below, the notation a ≈ b ( a, b >
0) denotes C − a ≤ b ≤ Ca . Lemma 4.3.
Let T ( x ) = (cid:18) e πiθ t ( x )0 e − πiθ (cid:19) , where t has a single non-zero Fourier coefficient, t ( x ) = ˆ t r e πirx . Let X = P kj =1 T ∗ j − T j − . Then (4.19) k X k ≈ k (1 + | ˆ t r | min { k , k θ − rα k − R / Z } ) , (4.20) k X − k − ≈ k. Proof.
We can compute explicitly T j = (cid:18) e πijθ t j e − πijθ (cid:19) with t j ( x ) = ˆ t r e πi ( rx +( j − θ ) e πijδ − e πiδ − , where δ = rα − θ . Write X = (cid:18) k x x x (cid:19) . By a straightforward computation,(4.21) x = ˆ t r e πi ( rx − θ ) k X j =1 e πi (2 j − δ − e πiδ − t r e πiδ − e πi ( rx − θ ) (cid:18) e πiδ e πikδ − e πiδ − − k (cid:19) , (4.22) x = k + | ˆ t r | k X j =1 (cid:18) sin π (2 j − δ sin πδ (cid:19) = k (cid:18) | ˆ t r | | e πiδ − | (cid:18) − sin 4 πkδ k sin 2 πδ (cid:19)(cid:19) . From those formulas we conclude that(4.23) det X = k | ˆ t r | | e πiδ − | − (cid:18) sin 2 πkδk sin 2 πδ (cid:19) !! . We first estimate x . If k k θ − rα k R / Z ≥ then1 − sin 4 πkδ k sin 2 πδ ≈ x ≈ k (1 + | ˆ t r | k θ − rα k R / Z ) . If k k θ − rα k R / Z < then 1 − sin 4 πkδ k sin 2 πδ ≈ ( k k θ − rα k R / Z ) , and x ≈ k (1 + k | ˆ t r | ). Since X is positive, we have kx ≥ | x | , and since x ≥ k we have x ≥ max { k, x } . Thus k X k ≈ x , and the first estimate follows.We now estimate det X . For k = 1 we have det X = 1. Assume that k >
1. If k k θ − rα k R / Z ≥ then 1 − (cid:18) sin 2 πkδk sin 2 πδ (cid:19) ≈ X ≈ k (1 + | ˆ t r | k θ − rα k R / Z ) . If k k θ − rα k R / Z < then 1 − (cid:18) sin 2 πkδk sin 2 πδ (cid:19) ≈ ( k k θ − rα k R / Z ) , and det X ≈ k (1 + k | ˆ t r | ) . This implies the second estimate, using that k X − k − = det X k X k . (cid:3) ONTINUITY OF SPECTRAL MEASURES 13
Lemma 4.4.
Let T , t and X be as in the previous lemma. Let ˜ T : R / Z → SL(2 , C ) . Let ˜ X = P kj =1 ˜ T ∗ j − ˜ T j − . Then (4.24) k ˜ X − X k ≤ , provided that k ˜ T − T k ≤ ck − (1 + 2 k k t k ) − . Proof.
Notice that k T j k ≤ j k t k , and(4.25) k ˜ T j − T j k ≤ j X s =1 (cid:18) js (cid:19) k ˜ T − T k s max ≤ i ≤ j k T i k s . Thus, if k ˜ T − T k k (1 + 2 k k t k ) is small we have k ˜ T j − T j k ≤ ck − (1 + 2 k k t k ) − , ≤ j ≤ k − . This implies k ˜ T ∗ j ˜ T j − T ∗ j T j k ≤ ck − , ≤ j ≤ k − , which gives the estimate. (cid:3) Theorem 4.5.
Let n = | n j | + 1 < ∞ , N = | n j +1 | . Then (4.26) k P ( k ) kk P − k ) k − ≤ C, Cn C < k < ce cN . Proof.
Let Φ, q , q , q , q be as in the definition of ( C , c , ǫ )-goodness.Let ∆ > n . Let | r | ≤ ∆ minimize k θ − rα k R / Z . Then | r | ≥ n −
1. By the Diophantine condition,(4.27) k θ − jα k R / Z ≥ c max { | r | , | j |} − C , for j = r such that | j | ≤ ∆ . Decompose q = t + g + h so that t has only the Fourier mode r , g has only the Fourier modes j = r with | j | ≤ ∆ and h is the rest. Then(4.28) Φ( x + α ) A ( x )Φ( x ) − = T + G + H, where T = (cid:18) e πiθ t e − πiθ (cid:19) , G = (cid:18) g (cid:19) , H = (cid:18) q hq q (cid:19) . By (3.4) and (3.5),(4.29) k H k ≤ Ce − cn − C ∆ + Ce − cN . Let Y = (cid:18) y (cid:19) be such that Y ( x + α )( T + G )( x ) Y ( x ) − = T ( x ) . Then we have(4.30) ˆ y j = − ˆ q j e − πiθ − e − πi (2 θ − jα ) , for j = r such that | j | ≤ ∆ , and ˆ y j = 0 if | j | > ∆ or j = r . Thus, by (3.5) and (4.27),(4.31) k Y − id k = k y k ≤ X j ≤ C (1+ | r | ) C | ˆ y j | + X C (1+ | r | ) C
For k ≥ , we have k P ( k ) k ≤ C k ( P ( k ) ) − k − .Proof. It follows from Theorem 4.5 and Lemma 3.1. (cid:3)
ONTINUITY OF SPECTRAL MEASURES 15
Set ǫ k = q
14 det P ( k ) . Corollary 4.7.
We have ψ ( m + ( E + iǫ k )) ≤ Cǫ − / k Proof.
We have k P ( k ) k = det P ( k ) k ( P ( k ) ) − k < Cǫ k k P ( k ) k − / . Thus k P ( k ) k ≤ Cǫ − / k and the state-ment follows from (4.11). (cid:3) Proof of Theorem 4.1.
By (4.4) it is enough to show that(4.43) cǫ / ≤ ℑ M ( E + iǫ ) ≤ Cǫ − / . For any bounded potential and any solution u we have k u k L +1 ≤ C k u k L , thus by (4.17), ǫ k +1 >cǫ k . Since ǫ ℑ M ( E + iǫ ) is monotonic in ǫ it therefore suffices to prove (4.43) for ǫ = ǫ k . But itfollows immediately from Corollary 4.7 and (4.6),(4.9). (cid:3)
Remark . Corollary 4.7 can be refined: for any 0 < ǫ < ψ ( m + ( E + iǫ )) ≤ Cǫ − / (thus it is not necessary to restrict to a subsequence of the ǫ k ). Indeed, by the definition of ψ ( z )it is enough to show that | m + β | ≤ Cǫ / for arbitrary β . Our proof can be easily adapted to show1 / µ + β associated to half-line problems (with appropriateboundary conditions) whose Borel transform is m + β , so that | m + β ( E + iǫ ) | ≤ Z dµ + β p ( x − E ) + ǫ (4.44) = Z ǫ dt µ ( x ∈ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | x − E | < r t − ǫ )! < Cǫ − / Z ∞ dx x / ( x + 1) / . Remark . It would be interesting to obtain estimates on the modulus of absolute continuity ofthe spectral measures. It does not seem unreasonable that for all
X, µ ( X ) ≤ C | X | / . Heuristically,the densities of the spectral measures are unbounded just because of the presence of those countablymany (but quickly decaying) square-root singularities located at the gap boundaries. We point outthat this is extremely similar to what is expected from the densities of physical measures of typicalchaotic unimodal maps.
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