Continuity of the spectrum of quasi-periodic Schrödinger operators with finitely differentiable potentials

Abstract

In this paper, we consider the spectrum of discrete quasi-periodic Schrödinger operators on $\ell ^{2}(\mathbb{Z})$ with the potentials $v\in C^{k}(\mathbb{T})$. For sufficiently large $k$, we show that the Lebesgue measure of the spectrum at irrational frequencies is the limit of the Lebesgue measure of the spectrum of its periodic approximants. This gives a partial answer to the problem proposed in Jitomirskaya and Mavi [Continuity of the measure of the spectrum for quasiperiodic schrödinger operator with rough potentials. Comm. Math. Phys.325 (2014), 585–601]. Our results are based on a generalization of the rigidity theorem in Avila and Krikorian [Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. of Math. (2)164 (2006), 911–940]; more precisely, we prove that in the $C^{k}$ case, for almost every frequency $\unicode[STIX]{x1D6FC}\in \mathbb{R}\backslash \mathbb{Q}$ and for almost every $E$, the corresponding quasi-periodic Schrödinger cocycles are either reducible or non-uniformly hyperbolic.

Ballistic Transport for One‐Dimensional Quasiperiodic Schrödinger Operators

“…(2) In [54], some of the results [9,10] were extended to the case of finitely smooth cocycles. As a consequence, the results from the previous case also extend to finitely differentiable potentials.…”

On Last’s intersection spectrum conjecture

“…However, there is no such theory for quasi-periodic Schrödinger operators with smooth potentials. In this paper, we give a new method to prove Chambers' formula based on C k global reducibility theories which are originally developed in [2,3,5,6,21]. Combining our Chambers' formula with a detailed classification of the spectrum, we get the final results.…”

“…Within the above concepts, we can state the global reducibility result obtained in [21](see corollary 6.1 in [21]):…”

Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum

“…In the case of Schrödinger operators (i.e., for b = 1), the statement ( 5) was previously established in various degrees of generality in the regime of positive Lyapunov exponents [23,27,18] and, in all regimes (using [3]), for analytic [25] or sufficiently smooth [49] v.…”

“…Typically, proofs that work for b = 1 extend also to the case of non-vanishing b, that is nonsingular Jacobi matrices, and there is no reason to believe the results of [25,49] should be an exception. On the other hand, extending various Schrödinger results to the singular Jacobi case is technically non-trivial and adds a significant degree of complexity (e.g.…”