Special Session 161: DYNAMICS AND SPECTRUM OF QUASIPERIODIC SCHRODINGER OPERATORS

Random compositions of quasi-periodic cocycles

Cai Ao Soochow University, School of Mathematical Sciences Peoples Rep of China Co-Author(s):    Pedro Duarte and Silvius Klein

Abstract: Random compositions of quasi-periodic cocycles serve as natural extensions of random matrix products in the fields of dynamical systems, probability theory, and group theory. This talk will begin with the historical development of random matrix product theory, introduce the concept of random compositions of quasi-periodic cocycles, and present existing results in this area.

Effective parameters and exceptional integrability

Fernando Argentieri IMPA Brazil Co-Author(s):    Artur Avila

Abstract: We study the problem of exceptional analytic integrability arising from perturbations. Our technique is based on checking the effectiveness of the available parameters. For definiteness, we will illustrate the technique with the case of one-frequency Schrodinger operators. This is joint work with Artur Avila

Aspects of quasi-periodicity in continuum models

Simon Becker ETH Zurich Switzerland Co-Author(s):

Abstract: We review results in the analysis of continuum models of quasi-periodic magnetic Schroedinger operators, focusing on spectral and dynamical properties in regimes of physical interest. We then discuss some new directions for graphene-based systems.

Dynamical Lifshitz Tails

Iris Emilsdottir University of California Irvine USA Co-Author(s):

Abstract: We consider one-parameter families of random circle diffeomorphisms $(g_{E,\\bar{\\omega}})$ for which the unperturbed map $g_{0,\\bar{0}}$ has a parabolic fixed point, and the dependence on the parameter $E$ is monotone. Under reasonable assumptions, we show that the rotation number $\\rho(E)$ exhibits Lifshitz tails: doubly-exponential decay with exponent $-(2k-1)/2k$, $ \\lim_{E \\downarrow 0} \\frac{\\ln(-\\ln(\\rho(E) - \\rho(0)))}{\\ln(E)} = -\\frac{2k-1}{2k}. $ The exponent is determined by the passage time through a parabolic bottleneck. A full rotation requires on the order of $E^{-(2k-1)/2k}$ consecutive small perturbations, and the probability of such a streak decays exponentially as a function of its length. As a corollary, we provide a purely dynamical proof of the classical Lifshitz tails result for the one-dimensional Anderson model.

On Floquet and Fermi Isospectrality

Matthew Faust Michigan State University USA Co-Author(s):

Abstract: In this talk we will discuss some recent progress on regarding the study of Fermi and Floquet isospectrality for discrete periodic Schr\odinger operators. We focus our discussion on various recently proven rigidity theorems regarding Fermi and Floquet isospectrality for the discrete Laplacian on Z^d with nearest neighbor edges. This talk will focus on two recent works of the author, one with Brysiewicz and Liu, and another with Cobb and Kretschmer.

Generalized Aubry-Andre formula and continuity of the intersection spectrum of the Almost Mathieu operator

Anton Gorodetski UC Irvine USA Co-Author(s):    Victor Kleptsyn

Abstract: In our joint work with Victor Kleptsyn, we consider the spectrum of the Almost Mathieu operator (AMO) and show that the moments of the restriction of the Lebesgue measure to the intersection spectrum $\operatorname{Leb}\big|_{\Sigma_{\alpha,\lambda}}$ are polynomials in coupling $\lambda$ with coefficients that are trigonometric polynomials in frequency $\alpha$. The statement can be considered as a generalization of the Aubry-Andr\`e formula for the measure of the spectrum of AMO. As a corollary, we obtain that the restriction of the Lebesgue measure to the intersection spectrum that we denote by $\mu_{\alpha, \lambda}^-$ depends continuously on the parameters (frequency $\alpha$ and coupling $\lambda$) in weak-* topology. Moreover, we prove that the dependence is not just continuous but analytic in $\lambda$ and $C^{\infty}$ in $\alpha$ in a sense that an integral of an analytic test function $\varphi(x)$ with respect to $\mu_{\alpha, \lambda}^-$ has the same kind of dependence. As the analyticity of the test function is here required only in a neighborhood of the union spectrum $\Sigma^+_{\alpha,\lambda}$, this implies that the Lebesgue measure of the part of the spectrum $\Sigma_{\alpha,\lambda}$ that lies between two gaps depends analytically on the coupling constant $\lambda$ and $C^{\infty}$ on the frequency $\alpha$ in an open domain (away from the critical coupling $\lambda=1$) where these gaps do not bifurcate.

Localization and spectral gaps for quasiperiodic operators with monotone potentials

Ilya Kachkovskiy Michigan State University USA Co-Author(s):    Svetlana Jitomirskaya, Leonid Parnovski, Roman Shterenberg

Abstract: In this talk, we will discuss recent advances in perturbative Anderson localization for quasiperiodic Schr\odinger operators, as well as methods based on rank one perturbation theory that allows to predict existence of spectral gaps and Cantor structure in their spectra. Based on joint papers with Svetlana Jitomirskaya, Leonid Parnovski, Roman Shterenberg.

Continuity of intersection spectrum at rationals

Xianzhe Li University of California, Berkeley USA Co-Author(s):

Abstract: For periodic Schr\odinger operators $H_{v,\alpha,x}$ with analytic potentials and frequency $\alpha\in \mathbb{Q}$. Let $S_-$ be the intersection of the spectra over all phases $x$. We show that $S_-(\alpha)$ is the limit of $S_-(p/q)$ for a class of rational approximations $p/q \to \alpha$, except possibly on a set of Lebesgue measure zero. This settles a question originally raised by S. Jitomirskaya. This is a joint work with S. Jitomirskaya and L. Shao.

New universality classes associated to fractals

Milivoje Lukic Emory University USA Co-Author(s):    Benjamin Eichinger, Alexander Kheifets, Peter Yuditskii

Abstract: The local behavior of zeros of orthogonal polynomials depends on the local properties of the measure, and different universality classes correspond to different local behaviors. From a modern perspective, this is analyzed through the scaling limits of the Christoffel-Darboux kernel. Recent works have established necessary and sufficient conditions for the most studied universality classes, including bulk universality and hard-edge universality. In all these settings, the rescaled Christoffel-Darboux kernel converges to a single limit kernel. In this talk, we introduce universality classes in which there is not a single limit kernel, but rather an entire cycle of limit kernels. We show that this type of behavior arises naturally from a fractal nature of the measure of orthogonality.

Lower Bounds for Green`s Function Fractional Moments of Random Operators and Their Consequences

Rodrigo Matos PUC-RIO Brazil Co-Author(s):

Abstract: We present lower bounds for Green`s function fractional moments of discrete random Schroedinger operators. These bounds take different forms, depending on whether the operator is local or non-local, and hold in various regimes of disorder and energy. We also extract consequences for the quantum dynamics and for the eigenfunction correlators. In particular, we present regimes where the fractional Anderson model does not exhibit dynamical localization (finiteness of q-th moments of the averaged position operator for any given q ) even in the presence of pure point spectrum with polynomially decaying eigenfunctions. For the Anderson model, we show optimality of exponential decay of correlators at large disorder. Based on joint works with: 1) Constanza Rojas-Molina and Peter Hislop, and 2) Sergey Sergeev.

Exotic spectral phenomenon of non-self-adjoint quasi-periodic operators

Zhenfu Wang Nankai University Peoples Rep of China Co-Author(s):    Jiawei He, Xueyin Wang, Jiangong You, Qi Zhou

Abstract: We give a precise description of the spectrum for a class of non-self-adjoint quasi-periodic operators. As applications, we obtain several exotic spectral phenomena, including interval spectra, two-dimensional spectra, and cases in which the spectrum coincides with that of the free Laplacian.

hyperbolic geodesic flow insights and spectral theory for long-range operators

Disheng Xu Great Bay University Peoples Rep of China Co-Author(s):

Abstract: In this talk I will present the recent joint work with Zhenfu Wang and Qi Zhou. We study the long-range operator, in particular we show any finite-range perturbation of a subcritical almost Mathieu operator, with Diophantine frequency, retains purely absolutely continuous spectrum for all phases. My talk will focus on the part that partiallly inspired by the study of hyperbolic geodesic flows.