Continuity of the measure of the spectrum for quasiperiodic Schrödinger operators with rough potentials

Transcription

1 Continuity of the measure of the spectrum for quasiperiodic Schrödinger operators with rough potentials Svetlana Jitomirskaya and Rajinder Mavi August 4, 202 Abstract We study discrete quasiperiodic Schrödinger operators on l 2 (Z) with potentials defined by -Hölder functions. We prove a general statement that for > /2 and under the condition of positive Lyapunov exponents, measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants. An important ingredient in our analysis is a general result on uniformity of the upper Lyapunov exponent of strictly ergodic cocycles. Introduction Consider quasiperiodic operators acting on l 2 (Z) and given by: (h ω,θ ψ)(n) = ψ(n ) + ψ(n + ) + f(ωn + θ)ψ(n), n =...,, 0,,..., (.) where f(x) is a real-valued sampling function of period. Denote by S(ω, θ) the spectrum of h ω,θ. For rational α = p/q the spectrum consists of at most q intervals. Let S(ω) = θ R S (ω, θ). Note that for irrational ω the spectrum of H (as a set) is independent of θ (see, e.g., [0]), and therefore S(ω, θ) = S(ω). In this paper we study continuity of S(ω) and its measure upon rational approximation of ω, for rough sampling functions f. The last decade has seen an explosion of general results for operators (.) with analytic f, see e.g. [7, 5] and references therein, and by now even the global theory of such operators is well developed [2, 3]. There are very few complete results, however, beyond the analytic category. Indeed, not only the methods of the mentioned papers intrinsically require analyticity or at least Gevrey (e.g. the large deviation theorems), but it is essential for some results too. For example, continuity of the Lyapunov exponent [8], an important ingredient of many later developments, may not hold in the case of even C regularity [3] (see also [8]). In this paper we show that, under certain conditions, for the question of continuity in ω analyticity is not essential. Namely, in the regime of positive Lyapunov exponents, spectra of rational approximants converge a.e. to S(ω) for all f, even with very low regularity: Hölder-/2+ continuity is sufficient. To our knowledge, other than the very basic facts that require, at most, continuity of f, there are no other results that do not require exclusion of potentially relevant parameteres or additional assumptions (e.g. transversality) and work for potentials that rough, particularly beyond the Lipshitz condition. The fact that various quantities could be easier to analyze and sometimes are even computable for periodic operators, H p/q,θ, makes results on continuity in ω particularly important. For example, the famous Hofstadter butterfly [3] is a plot of the almost Mathieu spectra for 50 rational values of ω and visually based inferences about the spectrum for irrational ω implicitly assume continuity. It is therefore an important and natural question if and in what sense the spectral properties of such rational approximants relate to those of the quasi-periodic operator H ω,θ. The work was supported by NSF Grant DMS-0578.

2 The history of this question was centered around the Aubry conjecture on the measure of the spectrum [], popularized by B. Simon [28, 29] : that for the almost Mathieu operator given by (.) with f(θ) = 2λ cos 2πθ, for irrational ω and all real λ, θ there is an equality S λ (ω, θ) = 4 λ. (.2) Here, for sets, we use to denote the Lebesgue measure. Avron,van Mouche, Simon [5] proved that, for λ, S λ (p n / ) 4 λ as, and Last [26] established this fact for λ =. Given these theorems, the proof of the Aubry-Andre conjecture was reduced to a continuity result. The continuity in ω of S(ω) in Hausdorff metric was proven in [6, ]. Continuity of the measure of the spectrum is a more delicate issue, since, in particular, S(ω) can be (and is, for the almost Mathieu operator) discontinuous at rational ω. We will actually use an even stronger notion of a.e. setwise continuity. Namely, we say lim n B n = B if and only if lim sup B n = lim inf B n = B lim χ B n = χ B Lebesgue a.e. (.3) n n n. Establishing continuity at irrational ω requires quantitative estimates on the Hausdorff continuity of the spectrum. The first such result, namely the Hölder- 3 continuity was proved in [9], where it was used to establish a zero-measure spectrum (and therefore the Aubry-Andre conjecture) for the almost Mathieu operator with Liouville frequencies ω at the critical coupling λ =. That argument was improved to the Hölder-/2 continuity (for arbitrary f C ) in [5] and subsequently used in [25, 26] to establish (.2) for the almost Mathieu operator for ω with unbounded continuous fraction expansion, therefore proving the Aubry-Andre conjecture for a.e. (but not all) ω. The extension to all irrational ω is due to [6, 4]. It was argued in [5] that Hölder continuity of any order larger than /2 would imply the desired continuity property of the measure of the spectrum for all ω. It was first noted in [20] that in the regime of semiuniform localization, the appropriate cut-offs of the exponentially localized eigenfunctions provide good enough approximate eigenvectors for a perturbed operator to establish almost Lipshitz continuity (thus establishing the Aubry-Andre conjecture in the localization regime available at that time). The idea of [6] was that for Diophantine ω and analytic f one can extract such eigenvectors (and thus establish almost Lipshitz continuity of S) by finding the cut-off places at distance L from each other where the generalized eigenfunction is exponentially small in L, simply as a corollary of positive Lyapunov exponents, without establishing localization. This led to establishing that, in the regime of positive Lyapunov exponents, for any analytic f, S( pn ) S(ω) for every Diophantine ω and its approximants pn. Recently, it was shown in [9] that positivity of the Lyapunov exponent is not needed for this result, in particular, for analytic f, and all irrational ω, S( pn ) S(ω). Our goal is to show that bringing back the condition of positivity of the Lyapunov exponent, allows to significantly relax the required regularity of f. For a given energy E R, a formal solution u of hu = Eu (.4) with operator h given by (.) can be reconstructed from its values at two consecutive points via ( ) ( ) u(n + ) = A E u(n) (θ + nω). (.5) u(n) u(n ) The function A E (θ) = ( E f(θ) 0 ) (.6) It should be noted that the argument of [4] that, in particular, completed the result for the critical value of λ, did not involve continuity in frequency 2

3 maps T into SL 2 (R). Setting R : T T, Rx := x + ω, the pair (ω, A E ) viewed as a linear skew-product (x, v) (Rx, A(x)v), x T, v R 2, is the called the corresponding Schrödinger cocycle. The iterations of the cocycle A for k 0 are given by and A E k (θ) = A E (R (k ) θ) A E (R θ)a E (θ), A 0 = I. (.7) A E k (θ) = ( A E k(r k θ) ) ; k < 0. (.8) Therefore, it can be seen from (.5) that a solution to (.4) for chosen initial conditions (u(0), u( )) for all k Z is given by, ( ) ( ) u(k) u(0) = A u(k ) k (θ, E). (.9) u( ) From general properties of subadditive ergodic cocycles, we can define the Lyapunov exponent L(E) = lim ln A E k (θ) dθ = inf ln A k (θ) dθ, (.0) k k k k furthermore, L(E) = lim k k ln AE k (θ) for almost all θ T. As mentioned above, L may be discontinuous in the non-analytic category. Set L + (ω) := {E : L(E) > 0}. Our main result is Theorem. For every irrational ω, there exists a sequence of rationals pn ω such that for any f C (T) with > 2 ( ) pn S L + (ω) S (ω) L + (ω). (.) Remark. The convergence holds in the strong sense of (.3). 2. The sequence pn will be the full sequence of continued fraction approximants of ω in the Diophantine case, and an appropriate subsequence of ( it otherwise. ) For practical purposes of making conclusions about S(α) based on the information on S pn it is sufficient to have convergence along a subsequence. 3. It is an interesting question whether = /2 represents a sharp regularity threshold for this result for a.e. ω. 4. Lower regularity is sufficient for a measure zero set of non-diophantine ω, see Theorem It is also interesting to find out what is the lowest regularity requirement for the convergence of full union spectra, without condition of positive L, and for the related Last s intersection spectrum conjecture. Both are more delicate and currently established only for analytic f ([9], see also [27]). We expect that higher than /2 regularity should be required for those results, but that analytic condition is improvable. 6. If we replace f in (.) with λf, then L is expected to be positive for most f and large λ through most of the spectrum, creating a wide range of applicability for Theorem.. For analytic f this is known to hold uniformly in (E, ω) f or large λ. For the rough case, the relevant results are [30, 2] Theorem. certainly has an immediate corollary: Corollary.2 For every irrational ω, there exists a sequence of rationals pn ω such that for any f C (T) with > 2 ( ) pn S L + (ω) S (ω) L + (ω). (.2) 3

4 This corollary for analytic f was the main result of [6] and our proof borrows some important ingredients from that work. The main idea of the current paper is to show that, for Diophantine frequencies, -Hölder continuity of f is sufficient to find the cut-off places at distance L from each other where the generalized eigenfunction is polynomially small in L, thus establishing β-hölder continuity of the spectrum with β <. The requirement > /2 comes from the application of the original argument in [5]. For non-diophantine frequencies we obtain the statement by extending the Hölder continuity theorem of [5] in the following way: + for -Hölder functions f the spectrum is -Hölder continuous, which is sufficient, under an appropriate anti-diophantine condition, even without positivity of the Lyapunov exponent, see Theorem 5.2. The proof requires very tight control on the perturbations of cocycles, in absence of continuity of the Lyapunov exponent. To this end, we show that generally, for cocycles over uniquely ergodic dynamics, upper bound is uniform in phases and neighborhoods (Theorem 3.2), a theorem that extends a uniformity result of Furman [2] and should be applicable to a variety of questions. The main part of the proof of Theorem. follows from Hölder continuity properties of the spectrum in the Hausdorff metric which are stated in section 5. The argument for the positive Lyapunov exponent regime uses tight bounds on matrix cocycle approximation covered in Section 4 which in turn depend on a general result on uniform upper-semicontinuity of Lyapunov exponents for cocycles over uniquely ergodic dynamics, that may be of independent interest, and is proven in Section 3. Section 6 completes the proof of Theorem.. 2 Continued fraction approximants For κ 0, ω R is said to be κ-diophantine if there exists some C ω > 0 so that nω > C n +κ (2.) for all n Z, where denotes distance to the integers. For κ > 0 a.e. ω T is κ-diophantine. For κ = 0 this condition is equivalent to having bounded type, so a.e. ω T is not 0-Diophantine. Writing ω in continued fraction expansion, ω = a 0 + a + a 2 + a = [a 0 ; a, a 2,...], the truncations p n / = [a 0 ; a, a 2,..., a n ] are known as the continued fraction approximants. From the theory of continued fractions [24], for κ-diophantine ω and n > n ω we have for some C ω > 0, C ω qn 2+κ < ω p n <. (2.2) + We will also need the following fact: Lemma 2. (e.g. [7]) For an interval I T, if n is such that I > then for any θ T there is 0 j + so that θ + jω I. We are now ready to formulate a more detailed version of the main Theorem Theorem 2.2 Assume f C (T) with > 0. Then. If ω is κ-diophantine, κ > 0, and > 2, then ( ) pn S L + (ω) S (ω) L + (ω). (2.3) for p n / the sequence of continued fraction approximants of ω. q 2 n 4

5 2. If ω is not κ-diophantine with κ =, then ( ) pn S S (ω) (2.4) for a subsequence of approximants Remark. Thus, for Lipshitz f (2.4) holds for a.e. ω (all except possibly for the bounded type). This is already implicit in [25]. 2. Theorem 2.2 certainly implies Theorem. 3 Uniform upper semicontinuity of the upper Lyapunov exponent This section is devoted to some fundamental properties of the Lyapunov exponent in the general setting. It is well known that the Lyapunov exponent of ergodic cocycles is upper semicontinuous. For a uniquely ergodic underlying dynamics, Furman [2] has shown, by an elegant subadditivity argument originally used by Katznelson and Weiss [23] to prove Kingman s ergodic theorem, that rate of convergence of a cocycle from above can be bounded uniformly in the phase. Now we investigate the coincidence of these properties. Assume (X, T, µ) is an ergodic Borel probability space. We use the notation {f} for a sequence (f n ) C(X, R) L (X, µ) which is a continuous subadditive cocycle with respect to T, that is f n+m (x) f n (x) + f m (T n x). The category of continuous subadditive cocycles will be denoted Γ(X). We define the Lyapunov exponent as, Λ(f) = lim f n (x)µ(dx). n n X By Kingman s subadditive ergodic theorem we have, for almost all x X, lim n n f n(x) = inf f n (x)µ(dx) = Λ(f). (3.) n n X To proceed it will be useful to introduce a metric on Γ(X). For two continuous cocycles {g}, {f} Γ(X) define d ({g}, {f}) = n 2 n g n f n 0 + g n f n 0. Then (Γ(X), d) is a metric topology. Since for any n the map {f} n follows that the infimum {f} inf f n = Λ(f) n n X X f n is continuous in (Γ(X), d), it is upper semicontinuous in (Γ(X), d). 2 On the other hand, for a fixed cocycle over uniquely ergodic dynamics the convergence is uniform in the phase. Theorem 3. (Furman [2]) Let {f} be a continuous subadditive cocycle on a compact uniquely ergodic space (X, T, µ). Given ɛ > 0, there exists n ɛ so that for n > n ɛ any x X has n f n(x) < Λ(f) + ɛ. 2 This is true for general L cocycles, with no continuity required. 5

6 We will combine these properties to obtain uniform uppersemicontinuity in both the cocycle and phase. We now inspect in further detail the convergence for any initial phase. Theorem 3.2 Let (X, T, µ) be a compact uniquely ergodic dynamical system (X, µ, T ). Then Λ : Γ(X) R is uniformly in x upper semicontinuous with respect to d. Proof We are to prove that given ɛ there exist δ ɛ, n ɛ such that for g with d(f, g) < δ ɛ and n > n ɛ, for all x X, n g n(x) Λ(f) + ɛ. Given small ɛ > 0, for x X let n(x) = inf{n N f n (x) < n (Λ(f) + ɛ)} be the minimum time the cocycle {f n } arrives near the Lyapunov exponent. Furthermore, define the open sets N A N := {x X n(x) N} = {x X f n (x) < n (Λ(f) + ɛ)}. n= Let N ɛ be large enough that, N N ɛ implies µ (A N ) > ɛ. Fix some N > N ɛ for the rest of the proof and let { N ɛ = {g} : d({f}, {g}) < ɛ } 2 N+. This implies, for x A N and g N ɛ, that In other words, if we define the set g n (x) < f n (x) + ɛ < n(x) (Λ(f) + 2ɛ). (3.2) B K = K {x X g n (x) n (Λ(f) + 2ɛ)}, n= then B K A K for K N and g N ɛ. For any x X construct a sequence (x i ) in X in the following way. subsequent terms let x i+ = T ni x i ; where n i is defined as For i = let x = x and for n i = n i (x) = { n(xi ), if x i A N, otherwise. Notice that n j N. We now consider the cocycles for a sufficiently large index. Let Q = sup{ g 0 : g N ɛ }, note if ɛ is small then Q < f 0 + 4ɛ. Let M > NQ ɛ, and choose p so that n + + n p M < n + + n p. Let K = M (n + + n p ) N. For g N ɛ, by subadditivity, p p g M (x) g ni (x i ) + g K (x p ) g ni (x i ) + NQ. i= We use the definition of n i and inequality (3.2), p [ g M (x) ni (Λ(f) + 2ɛ) AN (x i ) + Q X\AN (x i ) ] + NQ. i= 6 i=

7 Note, X\A N is a closed set of µ measure less than ɛ. By regularity of the Borel measure, there is an open set D containing X\A N of measure less than 2ɛ, and by Urysohn s lemma there is a continuous function g so that g X\AN = and g D c = 0. Therefore, for large M we have uniformly in x, p Q X\AN (x i ) M M i= Substituting this into the complete sum, M Q X\AN (T i x) M i= M Qg(T i x) < 3Qɛ i= M g M (x) p n i (Λ(f) + 2ɛ) AN (x i ) + M M i= M i= Q X\AN (T i x) + NQ M (3.3) (Λ(f) + 2ɛ) + 3Qɛ + NQ M (3.4) Λ(f) + 6Qɛ. (3.5) 4 Rate of convergence for matrix cocycles The first application of Theorem 3.2 is to approximations of matrix cocycles. Consider a continuous matrix A C (X, GL n (C)) defined on a compact uniquely ergodic space (X, µ, T ). Let the metric on C (X, GL n (C)) be defined by the norm A 0 = max θ A(θ). Then ln A n (θ) := ln A(T n x) A(x), A 0 = I, is a subadditive cocycle and its Lyapunov exponent is defined by L(A) = inf ln A(T n x) A(x). n n X Immediately, an application of Theorem 3.2 results in uniform uppersemicontinuity of the Lyapunov exponent: given ɛ, for D near A and large k, we have D k (x) exp{k(l(a) + ɛ)} (4.) uniformly in x, since D in a small C 0 neighborhood of A implies {ln D n } in a small d beighborhood of {ln A n }. This observation leads to the following Corollary 4. Let ɛ > 0, and A C (X, GL n (C)). For small enough δ and large k ɛ, if D A 0 < δ and k k ɛ, then A k D k 0 δe {k(l(a)+ɛ)}. (4.2) Proof By iterating the calculation, A k (θ) = A(T k θ) A(T θ) A(θ) = D(T k θ) A k (θ) + (A D) (T k θ) A k (θ) (4.3) we obtain the error bound A k (θ) D k (θ) D l (T k l θ)(d A)(T k l θ)a k l (θ) 0 l k 0 l k D l 0 D A 0 A k l 0. (4.4) 7

8 Now we invoke Theorem 3.2 for 0 < ɛ < ɛ to obtain δ > 0 and k(ɛ ) large so that D A 0 < δ and k > 2k(ɛ ) implies A k (θ) D k (θ) + + D l 0 D A 0 A k l 0 (4.5) δ 0 l k ɛ k ɛ l k k ɛ 2 δe {(k )(L+ɛ )} k + 2 k ɛ l k k ɛ 2 k k ɛ l k D l 0 A k l 0 + 2δe {(k )(L+ɛ )} 0 l k ɛ ( A + δ) l (4.6) 0 l k ɛ ( A + δ) l (4.7) δe {k(l+ɛ)} (4.8) for large enough k > k ɛ. Remark A standard argument would easily obtain (4.2) with exp{k(l + ɛ)} replaced by C A k 0. The issue here is tight control on the exponential rate of growth of the error, without assuming continuity of L. 5 Hölder Continuity in Frequency If I = [u, v] Z we write h I;θ = h [u,v],θ := R I h θ R I where R I projects onto the subspace of coordinates restricted to I. The Green s function for the interval is the inverse of the restriction G I (i, j) = δi T h I δ j. The determinants of the truncated matrix will be labeled Pk E(θ) := det(h [0,k ];θ E). The truncated Hamiltonian relates to the cocycle matrices by the equation [ P A E E k (θ) = k (θ) Pk E (θ + ω) Pk E (θ) P k E (θ + ω) ]. (5.) The following simple lemma allows to bound P k from above uniformly in θ and for a large measure subset of the spectrum Lemma 5. For any ζ, η > 0 there exists a set F (ζ, η) S(ω), F (δ, η) < ζ, and k(ω, ζ, η) = k F so that E S(ω)\F (ζ, η) and k > k F implies Pk E (θ) < e k(l(e)+η). (5.2) Furthermore there is some δ F > 0 so that uniform upper convergence in the sense of Lemma 4. holds. Thus, E S\F (ζ, η) implies if D A E < δ F and k > k F then (4.2) holds. Proof For all E there exists k E,η and δ E so that Corollary 4. holds. Thus, and Therefore, {E : k E,η > k} 0 as k {E : δ E < δ} 0 as δ 0. F (ζ, η) = {E : δ E < δ} {E : k E,η > k} for small enough δ and large enough k = k F so that F (ζ, η) < ζ. 8

9 5. The general case Here we observe that a result of Avron, Mouche and Simon on /2-Hölder continuity of the spectrum easily generalizes from f C to -Hölder case. Theorem 5.2 Suppose f C (T), > 0. Then E S(ω) and for small enough ω ω, there exists an E S(ω ) so that E E < C f ω ω +, for some constant Cf > 0 not depending on ω or ω. Note that by C f we mean a constant that depends only on f. Different such constants are denoted by the same C f in the proofs below. The proof is very similar to that of [5]. Starting with an approximate eigenfunction for h ω,θ E and using the same test function as in [5], upon a cutoff at a distance L we obtain an approximate eigenfunction for h ω,θ with an error in the kinetic energy of order L. The main difference is that the potential energy error is now bounded by CL ω ω, so the choice of L is optimized by L = C f ω ω +. More precisely, given ɛ > 0 and E S(ω), there exists an approximate eigenfunction φ ɛ l 2 (Z) so that ( ) + (h α,θ E)φ ɛ < ɛ φ ɛ. Set g j,l (n) = j n L, where g + (n) = g(n), n 0 and g + (n) = 0 otherwise. Avron-van Mouche-Simon [5] prove that for sufficiently large L for any bounded f : T R there exists j such that g j,l φ ɛ 0,and for any ɛ > 0, (h ω,θ E)g j,l φ ɛ 2 C ( ɛ 2 + L 2) g j,l φ ɛ 2, (5.3) where C is universal. Now let θ be given by ωj + θ = ω j + θ. By the Hölder assumption on f and j L n j + L, observe that Thus, f(θ + nω) f(θ + nω ) C f (L ω ω ) (h ω,θ E)g j,lφ ɛ (h ω,θ h ω,θ)g j,l φ ɛ + (h ω,θ E)g j,l φ ɛ ( C f (L ω ω ) + C ( ɛ 2 + L 2) ) /2 g j,l φ ɛ. (5.4) Since ɛ can be arbitrarily small, choosing L = C f ω ω +, to make both addends on the right-hand side of (5.4) equal, we obtain the statement of Theorem 5.2 by the variational principle. 5.2 Diophantine case As mentioned in [5] for Diophantine rotations /2-Hölder continuity of the spectrum (the best that can be obtained from Theorem 5.2) is not sufficient, so that is what we aim to improve. Theorem 5.3 Suppose h ω,θ is an operator of the form (.) where f C, > 0, ω [0, ] is κ- Diophantine, κ > 0. Fix 0 < β <. Given ζ > 0 there is a B ζ, 0 < B ζ < ζ so that for E S(ω) L + (ω)\b ζ and any ω near ω, there exists E S(ω ) such that E E < C f ω ω β. Remark The theorem holds for > β > 0, but the application we are interested in will require > β > 2. Proof We assume L + (ω) S(ω) otherwise the Theorem holds vacuously. Suppose f is -Hölder. Let 0 < β <. Let E χ = {E S(ω) : L(E) L + (ω) < χ}, with χ > 0 so small that E χ < ζ 2. By upper semicontinuity of the Lyapunov exponent, the Lyapunov exponent is bounded on compact sets. Let χ > 0 9

10 be an upper bound of the Lyapunov exponent on S(ω). Let > c > 3 4. Choose d so that c 2 > d > 4. Choose 0 < τ < ς < β d β + β ; > b > max( τ, c) and b < a <. (5.5) ( + 2κ) χ χ Finally, let η > 0 be such that 0 < η < min{χτ χ( b), χ( a), χ(c d )}. (5.6) 2 Define B ζ = E χ F (ζ/2, η) with F (, ) from Lemma 5. with associated k F and δ F. Take E S(ω) L + (ω)\b ζ.we now find an Nth degree trigonometric polynomial f N that approximates f. Namely, for - Hölder functions f, we have f N f < C f N where f N (θ) := K N f(θ) = K N being the Fejer s summability kernel, see for example [22]. Set N j N ( ( sin N+ 2 K N (θ) = θ) ) 2 sin ( 2 θ) = ( j ) ˆf(j)e ijθ, N + N j N { N = exp χk τ } ( j ) e ijθ. N + and let A (N),E be the cocycle matrix defined by the potential determined by the sampling function f N. For a map B : T SL 2 (R) and associated cocycle set { V k (t, B) = θ T : } k ln B k(θ) > t T. (5.7) The measure of this set for large enough k can be bounded below by use of Corollary 4.. Indeed, for k > k F we have for all θ, k ln AE k (θ) < L(E) + η, thus using (.0) and (4.), L(E) k ln ( AE k (θ) dθ V k al(e), A E ) (L(E) + η) + ( ( Vk al(e), A E ) ) al(e). (5.8) T Furthermore, we make the following claim regarding the sets V k (, ) for k > k G = max {k F, k a,b,c }, and E Ē, E Ē < exp{ χτk}, The left inclusion of (5.9), by (5.6), follows from A (N),Ē k V k (al(e), A E ) V k (bl(e), A (N),Ē) V k (cl(e), A Ē ). (5.9) ( θ V k al(e), A E ) = (θ) > e akl(e) C ( E Ē + N ) e k(l(e)+η) > e akl(e) Ce k(l(e)+η χτ) > e bkl(e). The right inclusion of (5.9) is similar, with comparisons (applications of Corollary 4.) made to A E, θ V k (bl(e), A (N),Ē) = A Ē k (θ) > e bkl(e) (CN + Ē Ē )e k(l(e)+η) > e bkl(e) Ce k(l(e)+η χτ) > e ckl(e), 0

11 again using (5.6). Using (5.9) and (5.8) we have V k (bl(e), A (N),Ē) χ χ + η/( a) 2. (5.0) Thus V k (bl(e), A (N),E ), being defined by a polynomial of order 4k exp{χkτ/}, contains an interval of length exp{ χkς/}, for sufficiently large k. It follows from (5.9) that V k (cl(e), AĒ) also contains an interval of length exp{ χkς/}. Now we move on to constructing the approximate eigenfunction. Let E 0 be a generalized eigenvalue of h ω,θ so that E E 0 < e ( χ+η)k, with generalized eigenvector ψ. By Sch nol s theorem, for spectrally a.e. E, ψ(x) = o(( + x ) /2+ɛ ), so we assume E 0 is such a value. Thus there exists an x m so that for all x Z. Normalize ψ so that, ψ(x m ) ( x m + ) ψ(x) ( x + ) ψ(x m ) ( x m + ) =. From (5.9) V k (cl(e), AĒ) contains an interval of length exp{ χkς/}. For any k sufficiently large there exists a denominator of an approximant, such that exp{kχς/} < exp {kχς( + 2κ)/}. (5.) Using Lemma 2., (5.9) and (5.) there exists an x, with x m 2 k x < x m k, A E0 k (T x θ) > e clk. Similarly, there exists x 3, with x m < x 3 x m + 2, such that A E0 k (T x 3 θ) > e cl(e)k. By (5.), for some k l, k r = k, k, or k 2 and x = x or x + and x 3 = x 3 or x 3 +, we have Let P E0 k l (T x θ), P E0 k r (T x3 θ) > 4 ecl(e)k. x l = x + [ ] kl ; x r = x [ ] kr. 2 Set also x 2 = x + k l and x 4 = x 3 + k r. Using Cramer s rule, as in [4] for ι = l, r and i =, 3 respectively, E0 G E0 [x (x P(x i,x i+] ι, x i ) < (T xι i+ x ι) θ) < C ( + exp { ( χ + η)k ι}) exp { } (L(E) + η) kι 2 < exp { dk ι L(E)} P E0 k ι (T xi θ) exp {cl(e)k ι } (5.2) with the numerator in the second inequality bounded above with (4.2) and the last inequality following from (5.6) for sufficiently large k; a similar statement holds for x ι = x l, x r + and i =, 3 respectively. Let { Λ = [x l, x r ] and let ψ Λ be the truncation of ψ to Λ or ψ Λ = R Λ ψ. We have Λ 4 +k < 5 exp by Lemma 2. and (5.). Now, by choice of x m, kχ ς(+2κ) } ψ(x ι ) x m + = ψ(x ι) x ι + x m + x ι + x ι + x m + x m + x ι x m + + x ι x m k/2 < 3 exp x m + { kχ } ς( + 2κ). (5.3)

12 Formal eigenfunctions ψ satisfy, for x x x 2, ψ(x) = G E0 [x,x 2] (x, x )ψ(x ) G E0 [x,x 2] (x, x 2)ψ(x 2 + ), (5.4) and similarly for x 3, x 4. Applying both (5.3) and (5.2) to (5.4) we obtain end point bound for Λ, { ψ(x l ), ψ(x l ), ψ(x r ), ψ(x r + ) C( x m + ) exp kχ The cutoff function then satisfies, (h ω,θ E 0 )ψ Λ C( x m + ) exp Define φ Λ = ψ Λ / ψ Λ. Now note that ψ Λ x m + so that (h ω,θ E 0 )φ Λ x m + (h ω E 0 )ψ Λ C exp ς( + 2κ) { ( )} ς( + 2κ) k dl(e) χ { k } exp { kdl(e)} (5.5) ( dl(e) χ For ω T set θ = θ xr+x l 2 (ω ω ). Then, perturbing the Hamiltonian s frequency, (h ω,θ h ω,θ ) φ Λ Thus )} ς( + 2κ). max f(θ +xω ) f(θ +xω) C f ( Λ ω ω ) < C f ω ω exp{kχς(+2κ)} x l x x r (5.6) (E h ω,θ ) φ Λ E E 0 + (E 0 h ω,θ )φ Λ + (h ω,θ h ω,θ ) φ Λ (5.7) { ( )} ς( + 2κ) E E 0 + C exp k dl(e) χ + C f ω ω exp{kχς( + 2κ)} { ( )} ς( + 2κ) C exp k dl(e) χ + C f ω ω exp{kχς( + 2κ)}. (5.8) Thus, by the variation principle, there exists an E in S(ω ) so that If we take k > k G such that E E (E h ω,θ ) φ Λ. (5.9) β ln ω ω ( ) k ( β) ln ω ω, χ d ς ( + 2κ) χς( + 2κ) which we can do, by (5.5), for sufficiently small ω ω, we obtain E E < ω ω β. The required smallness of ω ω depends only on chosen parameters, therefore on ω (through its Diophantine parameters), β, ζ and f. 6 The strong continuity. Proof of Theorem 2.2 This argument is very similar to that of [6] (which in turn is a modification of the proof in [25]). First, continuity of S + (α) in Hausdorff metric [5] implies ( ) p lim sup S + Σ(α), (6.) p q α q 2

13 for any irrational α T (inclusion holds set-wise, not just a.e, and for any continuous f), which immediately implies the corresponding inclusion in Theorem 2.2. For the opposite inclusion we need to consider continued fraction approximants pn. Note that because of continuity in θ, the set S + (p n / ) consists of at most disjoint intervals, say S + (p n / ) = q n i= [a n,i, b n,i ], q n. We now treat Diophantine and non-diophantine cases separately. For a Diophantine ω, Theorem 5.3 implies that for n > n(ω, β, ζ, f), thus S(ω) L + (ω) q n i= [a n,i C f ω p n β, b n,i + C f ω p n β ] B ζ (S(ω) L + (ω)\b ζ )\S + (p n / ) < 2C f ω p n β 0 since β > /2. Therefore, for every ζ > 0, we have S(ω) L + (ω)\b ζ )\ lim inf pn/ ω S + (p n / ) = 0. Thus S(ω) L + (ω)\ ζ>0 B ζ )\ lim inf p n/ ω S +(p n / ) = 0, which gives the desired inclusion in Theorem.. p Now, consider the irrational ω so that there exists a sequence of rational n qn so that p n and are mutually prime and + q n ω p n 0, (6.2) so that, ω is not κ Diophantine for κ >. Similar to the above calculation, we have, letting S ( pn qn ) = i q n [a n,i, b n,i ], and using Theorem 5.2 that S(ω) i [ a n,i C f ω p n +, bn,i + C f ω p n + ] Thus, by (6.2), S(ω) lim inf p n/ ω S ( pn ). References [] Aubry and A. Andre. Analyticity breaking and Anderson localization in incommensurate lattices. Annals of the Israeli physical society, 3:33 64, 980. [2] Artur Avila. Global theory of one-frequency Schrödinger operators I: Stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity. preprint, arxiv: [math.ds]. [3] Artur Avila. Global theory of one-frequency Schrödinger operators II: Acriticality and finiteness of phase transitions for typical potentials. preprint, 20. [4] Artur Avila and Rafaël Krikorian. Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles. Annals of Mathematics, 64:9 940, [5] Joseph Avron, Pierre Mouche, and Barry Simon. On the measure of the spectrum for the almost Mathieu operator. Communications in Mathematical Physics, 32:03 8,

14 [6] Joseph Avron and Barry Simon. Almost periodic Schrödinger operators II. The integrated density of states. Duke Math Journal, 50:369 38, 982. [7] Jean Bourgain. Green s function for lattice Schrödinger operators and applications. Princeton University Press, [8] Jean Bourgain and Svetlana Jitomirskaya. Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potentials. Journal of Statistical Physics, 08:203 28, [9] Man-Duen Choi, George A. Elliott, and Noriko Yui. Gauss polynomials and the rotation algebra. Inventiones Mathematicae, 99: , /BF [0] H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon. Schrödinger operators with applications to quantum mechanics and global geometry. Springer, 987. [] G.A. Elliot. Gaps in the spectrum of an almost periodic Schrödinger operator. Comptes Rendus Mathématiques. Acad Sci. Canada, 4: , 982. [2] Alex Furman. On the multiplicative ergodic theorem for uniquely ergodic systems. Probabilities et Statistiques, 33(6):797 85, 997. [3] Douglas R. Hofstadter. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B, 4: , Sep 976. [4] S. Y. Jitomirskaya. Metal-insulator transition for the almost Mathieu operator. Ann. of Math., 50:59 75, 999. [5] Svetlana Jitomirskaya. Ergodic Schrödinger operators (on one foot). In Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon s 60th birthday, pages Amer. Math. Soc., [6] Svetlana Jitomirskaya and I. V. Krasovsky. Continuity of the measure of the spectrum for discrete quasiperiodic operators. Mathematical Research Letters, 9(4):43 42, 200. [7] Svetlana Jitomirskaya and Yoram Last. Power law subordinacy and singular spectra. II. Line operators. Communications in Mathematical Physics, 2: , [8] Svetlana Jitomirskaya and Chris Marx. Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper s model. Communications in mathematical physics, 20. to appear. [9] Svetlana Jitomirskaya and Chris Marx. Analytic quasi-periodic Schrödinger operators and rational frequency approximants. GAFA, 202. to appear. [20] Svetlana Y. Jitomirskaya and Yoram Last. Anderson localization for the almost Mathieu equation, III. Semi-uniform localization, continuity of gaps, and measure of the spectrum. Communications in Mathematical Physics, 95: 4, 998. [2] K. Bjerklöv. Positive Lyapunov exponent and minimality for a class of -d quasi-periodic Schrödinger equations. Ergodic Theory Dynam. Systems, 25:05045, [22] Yitzhak Katznelson. Harmonic Analysis. Cambridge University Press, [23] Yitzhak Katznelson and Benjamin Weiss. A simple proof of some ergodic theorems. Israel Journal of Mathematics, 42:29 296, 982. [24] A. Ya. Khinchin. Continued Fractions. Dover, 964. [25] Y. Last. A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants. Communications in Mathematical Physics, 5:83 92,

15 [26] Yoram Last. Zero measure spectrum for the almost Mathieu operator. Communications in Mathematical Physics, 64:42 432, 994. [27] Mira Shamis. Some connections between almost periodic and periodic discrete Schrödinger operators with analytic potentials. Journal of Spectral Theory, : , 20. [28] Barry Simon. 5 problems in mathematical physics. In Perspectives in mathematics, pages Birkhäuser, 984. [29] Barry Simon. Schrödinger operators in the twentieth century. Journal of Mathematical Physics, 4(6): , [30] Thomas Spencer. Ergodic Schrödinger operators. Analysis, et cetera,, pages , 990. [3] Yiqian Wang and Jiangong Yu. Examples of discontinuity of Lyapunov exponent in smooth quasiperiodic cocycles. preprint, 202. arxiv: v2[math.ds]. 5