Optimality of log Hölder continuity of the integrated density of states
aa r X i v : . [ m a t h . SP ] J un OPTIMALITY OF LOG H ¨OLDER CONTINUITY OF THEINTEGRATED DENSITY OF STATES
ZHENG GAN AND HELGE KR ¨UGER
Abstract.
We construct examples, that log H¨older continuity of the inte-grated density of states cannot be improved. Our examples are limit-periodic. Introduction
We investigate optimality of the log H¨older continuity of the integrated densityof states. Let (Ω , µ ) be a probability space, T : Ω → Ω an invertible ergodictransformation, and f : Ω → R a bounded measurable function. Define a potential V ω ( n ) = f ( T n ω ). The Sch¨odinger operator H ω : ℓ ( Z ) → ℓ ( Z ) is defined by(1.1) H ω u ( n ) = u ( n + 1) + u ( n −
1) + V ω ( n ) u ( n ) , and the integrated density of states k by(1.2) k ( E ) = lim N →∞ Z Ω (cid:18) N tr( P ( −∞ ,E ) ( H ω, [0 ,N − )) (cid:19) dµ ( ω ) , where H ω, [0 ,N − denotes the restriction of H ω to ℓ ([0 , N − Thoulessformula , Craig and Simon showed that
Theorem 1.1 (Craig and Simon, [4]) . There exists a constant C = C ( k f k ∞ ) suchthat (1.3) | k ( E ) − k ( ˜ E ) | ≤ C log | E − ˜ E | − for | E − ˜ E | ≤ . This is what is well known as log H¨older continuity . We will be interested in theoptimality of this statement in the sense that ε ε − ) cannot be replaced byanother function, which goes to zero faster. It was shown by Craig in [3], that theregularity cannot be improved to ε ε − ) log(log( ε − )) β , where β >
1. However, in the case of specific dynamical systems (Ω , µ, T ), thereexist many results, which improve the Craig–Simon result. We just mention two.For quasi-periodic Schr¨odinger operators, Goldstein and Schlag have shown in [7]that the integrated density of states is H¨older continuous and computed the H¨olderexponent, and shown that the integrated density of states is almost everywhere
Date : October 29, 2018.2000
Mathematics Subject Classification.
Primary 47B36; Secondary 47B80, 81Q10.
Key words and phrases.
Integrated density of states, limit periodic potentials.H. K. was supported by NSF grant DMS–0800100.
Lipschitz. For random Schr¨odinger operators, the integrated density of states iseven everywhere Lipschitz. This is known as the Wegner estimate which can befound for example in the exposition of Kirsch in [6].Our interest in the question of optimality of the Craig–Simon results comes fromthe importance of the Wegner estimate in multiscale analysis (see the exposition ofKirsch). If one could improve the result to a continuity of the form ε ε − ) β for some large enough β >
1, one would be able to use this for multiscale analysis(see for example Theorem 3.12 in [8]). Already Craig’s result shows that this isimpossible, however one could hope that a combination of an improved continuityresult and an improvement of multiscale analysis might remove the Wegner estimateassumption. However, we will show that the continuity of integrated density ofstates cannot be improved for all potentials beyond log H¨older continuity.A potential V ∈ ℓ ∞ ( Z ) is called almost-periodic, if the closure Ω of its translatesis compact in the ℓ ∞ norm. Furthermore, then Ω can be made into a compactgroup, with an unique invariant Haar measure. For these our previous definition ofthe integrated density of states (1.2) can be replaced by(1.4) k V ( E ) = lim N →∞ N tr( P ( −∞ ,E ) ( H [0 ,N − )) , where H [0 ,N − denotes now the restriction of ∆ + V to ℓ ([0 , N − V is called p periodic, if its p -th translate is equal to V . Furthermore, V is limit-periodic if it is the limit in the ℓ ∞ norm of periodic potentials. We denoteby σ (∆ + V ) the spectrum of the operator ∆ + V . Theorem 1.2.
Given any increasing continuous function ϕ : R + → R + with (1.5) lim x → ϕ ( x ) = 0 and a constant C > , there is a limit-periodic V satisfying k V k ∞ ≤ C such thatits integrated density of states satisfies (1.6) lim sup E → E | k V ( E ) − k V ( E ) | log( | E − E | − ) ϕ ( | E − E | ) = ∞ , for any E ∈ σ (∆ + V ) . This result tells us, that with ϕ as in the previous theorem, we cannot have | k V ( E ) − k V ( E ) | ≤ C · ϕ ( | E − E | )log( | E − E | − )for any C > V . The proof of this theorem essentially happens in twoparts. Given a periodic V and ε > k V k ≤ C − ε , we construct asequence V j of periodic potentials, with the following properties(i) V j is p j -periodic.(ii) The Lebesgue measure of σ (∆ + V j )(1.7) ε j = | σ (∆ + V j ) | PTIMALITY OF LOG H ¨OLDER CONTINUITY OF THE IDS 3 satisfies(1.8) log( ε − j ) ≥ p j − · p j · ϕ (2 ε j ) . (iii) We have that(1.9) k V j − V j − k ≤ min( ε, ε , . . . , ε j − )2 j . Here k . k denotes the ℓ ∞ norm. The construction of these V j will be given in thenext section and uses the tools developed by Avila in [1]. Before proceeding withthe proof of Theorem 1.2, recall that positivity of the trace implies that k W ( E − k V − W k ) ≤ k V ( E ) ≤ k W ( E + k V − W k )for any potentials V and W . Proof of Theorem 1.2.
By (1.9), we see that there exists a limiting potential V ,such that for each j , we have that k V j − V k ≤ ε j = | σ (∆ + V j ) | . Furthermore, since k V − V k ≤ ε , we have that k V k ≤ C .Next, fix E ∈ σ (∆ + V ) and let j ≥
1. By the previous equation, we have thatthere exists E ∈ σ (∆ + V j ) such that | E − E | ≤ ε j = | σ (∆ + V j ) | . Denote the band of σ (∆ + V j ) containing E by [ E − , E + ]. By a general fact aboutperiodic Schr¨odinger operators, we know that k V j ( E + ) − k V j ( E − ) = 1 p j . We thus get that k V ( E + + ε j ) − k V ( E − − ε j ) ≥ p j Furthermore, the interval [ E − − ε j , E + + ε j ] contains E and we can choose E j ∈{ E − − ε j , E + + ε j } such that | k V ( E ) − k V ( E j ) | ≥ p j and | E − E j | ≤ ε j . This implies the claim by (1.8), since j was arbitrary. (cid:3) One can slightly improve the above theorem, by for example showing that thereis not only one V , that satisfies the conclusion, but that in fact the set is dense inthe limit-periodic operators. However, we have not done so, to keep the statementas simple as possible. Z. GAN AND H. KR¨UGER Construction of the periodic potentials
We will need the machinery developed by Avila in [1], in order to prove ourresults. In the following, we let Ω be a totally disconnected compact group, knownas
Cantor group . We furthermore let T : Ω → Ω be a minimal translation on thisgroup. There is a decreasing sequence of Cantor subgroups X ⊇ X ⊇ . . . such that the quotients Ω /X k contain p k elements. We let P k be the subset of the continuous functions C (Ω) onΩ, which only depend on Ω /X k . f is called n -periodic if f ( T n ω ) = f ( ω ) for every ω ∈ Ω. The elements of P k will be p k periodic.We now fix ω ∈ Ω. We have that { f ( T n ω ) } n ∈ Z ∈ ℓ ∞ ( Z ) is limit-periodic, sincethe periodic f are dense in C (Ω). For a finite subset F of the periodic potentials P = S k ≥ P k , we introduce the averaged Lyapunov exponent L ( E, F ) as(2.1) L ( E, F ) = 1 F X f ∈ F L ( E, f ) , where F denotes the number of elements of F (with multiplicities) and L ( E, f ) = lim N →∞ N log (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y n = N (cid:18) f ( T n ω ) − E −
11 0 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) is the Lyapunov exponent of the periodic potential. For f ∈ C (Ω), we denote byΣ( f ) the spectrum of the operator ∆ + f ( T n ω ). We will use the following twolemmas of Avila [1], see also [5]. Lemma 2.1 (Lemma 3.1. in [1]) . Let B be an open ball in C (Ω) , let F ⊂ P ∩ B be finite, and let < ε < . Then there exists a sequence F K ⊂ P ∩ B such that (i) L ( E, F K ) > whenever E ∈ R , (ii) L ( E, F K ) → L ( E, F ) uniformly on compacts. Lemma 2.2 (Lemma 3.2. in [1]) . Let B be an open ball in C (Ω) , and let F ⊂ P k ∩ B be a finite family of sampling functions. Then for every N ≥ and K sufficientlylarge, there exists F K ⊂ P K ∩ B such that (i) L ( E, F K ) → L ( E, F ) uniformly on compacts, (ii) The diameter of F K is at most p − K , (iii) For every λ ∈ R , if (2.2) inf E ∈ R L ( E, F ) ≥ δ F p k , then for every f ∈ F K , the spectrum Σ( f ) has Lebesgue measure at most e − δp K / . The construction of the V j will be accomplished by Proposition 2.3.
Given a continuous function ψ : R + → R + satisfying (2.3) lim x → ψ ( x ) = 0 , and p -periodic f and ε > , then there exists a ˜ p -periodic function ˜ f , such that (2.4) k f − ˜ f k ∞ ≤ ε, PTIMALITY OF LOG H ¨OLDER CONTINUITY OF THE IDS 5 and (2.5) log( | Σ( ˜ f ) | − ) ≥ ˜ p · ψ ( | Σ( ˜ f ) | ) . Proof.
By Lemma 2.1, we can find a finite family of p -periodic potentials F within B ε ( f ) such that δ = 1 F p L ( E, F ) = 1 F p · F X f ∈ F L ( E, f ) > . Applying Lemma 2.2 to F , we can get a finite family F ⊂ B ε ( f ) of ˜ p -periodicpotentials, where we might require ˜ p to be arbitrarily large. Let ˜ f be any elementof F . By Lemma 2.2 (iii), we have that | Σ( f ) | < e − δ ˜ p/ . Hence, (2.5) turns into 12 δ ˜ p ≥ ˜ pψ (e − δ ˜ p/ ) . The claim now follows from the fact, that ψ ( x ) → x → (cid:3) It now remains to construct the sequence of potentials V j . We proceed by in-duction. By possibly modifying (Ω , T ), we can assume that V ( n ) = f ( T n ω )for some f ∈ C (Ω). Assume now that, we are given V = f ◦ T n , . . . , V j − = f j − ◦ T n and we wish to construct V j = f j ◦ T n . We can choose now ε in the previousproposition to be the right hand side of (1.9). We choose ψ ( x ) = p j − ϕ (2 x ) andthe claim follows. References [1] A. Avila,
On the spectrum and Lyapunov exponent of limit periodic Schr¨odinger operators ,Comm. Math. Phys. (2009), 907–918[2] J. Avron, B. Simon,
Almost periodic Schr¨odinger operators II. The integrated density ofstates , Duke Math. J. (1983), 369–391.[3] W. Craig,
Pure point spectrum for discrete almost periodic Schr¨odinger operators , Comm.Math. Phys. (1983), 113–131.[4] W. Craig, B. Simon,
Subharmonicity of the Lyaponov index , Duke Math. J. (1983),551–560.[5] D. Damanik, Z. Gan,
Limit-periodic Schr¨odinger operators in the regime of positive Lya-punov exponents , preprint.[6] M. Disertori, W. Kirsch, A. Klein, F. Klopp, V. Rivasseau,
Random Schr¨odinger Operators ,Panoramas et Synthses (2008), xiv + 213 pages.[7] M. Goldstein, W. Schlag, Fine properties of the integrated density of states and a quan-titative separation property of the Dirichlet eigenvalues , Geom. Funct. Anal. (2008),755–869.[8] H. Kr¨uger,
Multiscale Analysis for Ergodic Schr¨odinger operators and positivity of Lyapunovexponents ,. preprint.
Department of Mathematics, Rice University, Houston, TX 77005, USA
E-mail address : [email protected] URL : http://math.rice.edu/ ∼ zg2/ Department of Mathematics, Rice University, Houston, TX 77005, USA
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