Special Session 164: Periodic and Ergodic Schrodinger Operators

Hidden Criticality and the Critical Dry Ten Martini Problem

Dan Borgnia University of California, Berkeley USA Co-Author(s):    Dan Borgnia, Robert-Jan Slager

Abstract: The almost Mathieu operator (AMO) famously hosts a 1D metal-insulator phase transition. Away from the transition, the AMO spectrum has been shown to form a Cantor set for all irrational frequencies (Ten Martini Problem) with all labeled gaps open (Dry Ten Martini Problem). This work closes problem for the remaining $\lambda =1$ critical point at Diophantine frequencies, showing the critical AMO Cantor set spectrum attains all labeled gaps. Our proof depends on two key ingredients: (1) Sub/Supercritical Embedding -- Hidden singularities of critical AMO in the chiral gauge and a finite similarity transform permit us to embed finite-volume restrictions of the critical AMO into the non-critical operator. (2) Bulk-Edge Correspondence -- The closing of labeled gaps at $\lambda =1$ forces small rank-one errors via bulk-edge correspondence and permit the gluing of the finite-volume restrictions back together. Together, these imply that if a labeled gap closes, the infinite critical operator may be glued together and fully embedded in any non-critical operator (at re-scaled energy) for which the non-critical AMO resolvent is lower bounded by a diverging critical AMO resolvent. This is too strong a condition and violates the AMO gap continuity, arriving at a contradiction.

Anderson localization for analytic quasi-periodic Schr\odinger operators

Hongyi Cao Peking University Peoples Rep of China Co-Author(s):    Yunfeng Shi, Zhifei Zhang

Abstract: In this talk, we discuss some recent progress on Anderson localization for quasi-periodic Schr\odinger operators on $\Z^d$ with analytic potentials and fixed Diophantine frequencies.

Weak separability and partial Fermi isospectrality of discrete periodic Schr\\odinger operators

Jifeng Chu Hangzhou Normal University Peoples Rep of China Co-Author(s):

Abstract: We consider the discrete periodic Schr\odinger operators $\Delta+V$ on $\Z^d$, where $V$ is $\Gamma$-periodic with $\Gamma=q_1 \mathbb{Z}\oplus q_2\mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$ and positive integers $q_j$, $j=1,2,\cdots,d$ are pairwise coprime. By introducing the notions of generalized partial Fermi isospectrality and weak separability, we prove that two generalized partially Fermi isospectral potentials have the same weak separability. As a direct application, we prove that two potentials have the same $(d_1,d_2,\cdots,d_r)$-separability by assuming that the projections of Fermi varieties or Bloch varieties on some $3$-dimensional subspace are the same, instead of the coincidence of the whole Fermi varieties or Bloch varieties. Besides, we prove that each couples of components of the generalized Fermi isospectral potentials are Floquet isospectral.

Sharp Polynomial Decay Bounds for Multidimensional Periodic Schroedinger Operators

Christoph Fischbacher Baylor University USA Co-Author(s):

Abstract: In this talk, I will present our results on discrete periodic Schroedinger operators in arbitrary dimensions in the large coupling regime. We establish that both the Lieb-Robinson velocity and the asymptotic velocity decay at an inverse polynomial rate in the coupling, with the precise exponent determined by the period of the underlying potetnial. In particular, we show sharp polynomial decay rates that capture the precise dependence on the periodic structure

The Fourier Ratio and Chang`s Lemma

Giovanni Garza University of Delaware USA Co-Author(s):    K. ALDALEH, W. BURSTEIN, G. GARZA, G. HART, A. IOSEVICH, J. IOSEVICH, A. KHALIL, J. KING, N. KULKARNI, T. LE, I. LI, A. MAYELI, B. MCDONALD, K. NGUYEN, AND N. SHAFFER

Abstract: Given a function $f:\Z_N\to\C$, we denote by $\widehat{f}$ its Fourier transform which is given by \begin{equation*} \widehat{f}(m)=\frac{1}{\sqrt{N}}\sum_{x\in\Z_N}e^{-2\pi ixm/N}f(x). \end{equation*} We introduce the function $FR(f)=\frac{\norm{\widehat{f}}_1}{\norm{\widehat{f}}_2}$, and examine how this ratio tells us the extent to which we can approximate $f$ by a low degree trigonometric polynomial. Finally, we prove a generalized version of Chang`s lemma and show that the sparse spectrum approximation of $f$ has some additive structure.

Moduli of Continuity of Lyapunov Exponents for Random Non-Invertible Cocycles

Tome Filipe Silvestre T. Graxinha Centro de Estudos Matematicos - Universidade de Lisboa Portugal Co-Author(s):    Pedro Duarte

Abstract: We study the quantitative regularity of Lyapunov exponents for non-compact and non-invertible random linear cocycles, with respect to the Wasserstein distance. Our first main result shows that, under three natural hypotheses (finite exponential moments, quasi-irreducibility, and a spectral gap $L_1 > L_2$) the top Lyapunov exponent depends locally H\{o}lder-continuously on the measure that governs the dynamics. The proof relies on a spectral method in which the strong mixing of the associated Markov operator is crucial. To that end, we extend Furstenberg-Kifer theory beyond the invertible setup. Consequences include H\{o}lder continuity of higher exponents via exterior powers, large-deviations--type estimates, and applications to Schr\{o}dinger cocycles with unbounded potentials. Our second main result shows that the exponential moment assumption is sharp. We construct non-compact random Schr\{o}dinger cocycles with locally uniform sub-exponential moments but without any exponential moment, for which the Lyapunov exponent, as a function of the energy, fails to be $\alpha$-H\{o}lder for every $\alpha > 0$. More generally, we provide a correspondence between moment profiles and moduli of continuity, and we prove that the breakdown of a given (locally uniform) moment condition prevents the corresponding modulus of continuity from holding.

Localization for the non-stationary Anderson model in two and three dimensions

Omar Hurtado Georgia Institute of Technology USA Co-Author(s):

Abstract: We discuss a non-stationary variant of the Anderson model, a celebrated and fundamental model in the study of disordered systems in condensed matter. We will discuss localization results for such models obtained in two and three dimensions by the speaker, and (time permitting) discuss the methods, including e.g. Bernoulli decompositions and unique continuation bounds.

Inverse Eigenvalue Problems, Floquet Isospectrality and the Hilbert--Chow Morphism

Andreas Kretschmer HU Berlin Germany Co-Author(s):

Abstract: When can we change the diagonal of a matrix without changing its spectrum? We answer this question as follows: A square matrix $A$ admits a nonzero diagonal matrix $D$ such that $A$ and $A+D$ have the same spectrum if and only if there are two principal minors of $A$ of the same size that are not equal. This relates to the classical additive inverse eigenvalue problem in numerical analysis and has implications for existence and rigidity results in the theory of Floquet isospectrality of discrete periodic Schr\{o}dinger operators. The proof employs new techniques involving Hilbert schemes of points and the infinitesimal structure of the Hilbert--Chow morphism. This is joint work with John Cobb and Matt Faust.

Ergodicity in discrete-time quantum walks

Kiran Kumar New York University Abu Dhabi United Arab Emirates Co-Author(s):    Mostafa Sabri

Abstract: In this talk, we present some results on ergodicity for homogeneous discrete-time quantum walks on the integer lattice. The most significant result holds in dimension one and gives a complete equivalence between the absolutely continuous spectrum of the unitary operator encoding the walk and the equidistribution of its dynamics in position space. In higher dimensions, we give a criterion for full and partial ergodicity in terms of a finer property of the spectrum which we dub `No Repeating Graphs`, and distinguish how strongly the equidistribution is taking place (weak convergence vs total variation). We also present applications of our results to the ergodicity of eigenvectors for Schr\{o}dinger operators on $\mathbb{Z}$-periodic graphs.

Singular Spectrum under a Wide Class of Perturbations

Constanze D Liaw University of Delaware USA Co-Author(s):    Eero Saksman, Sergei Treil

Abstract: For bounded self-adjoint operators $A$ and $K$ on a separable Hilbert space, consider perturbed operators of the form $A+K$. We present restrictions on the singular spectrum under trace class and more general perturbations. Some of the results are for a one-parameter family of perturbations $A+tK$, as the real parameter $t$ varies. To the best of our knowledge, these are the first statements on the singular spectrum under infinite rank perturbations.

Eigenvalues of Maximal Abelian Covers

Mostafa Sabri New York University Abu Dhabi United Arab Emirates Co-Author(s):    Wenbo Li, Michael Magee and Joe Thomas.

Abstract: I will give a quick overview of the problem of understanding flat bands (eigenvalues) of periodic graphs. I will then focus on maximal abelian covers, which form an important class of periodic graphs, and for which we obtained a complete characterization of the flat bands in terms of the combinatorics of the base graph. Coupling this criterion with a sophisticated analysis, we proved a conjecture of Higuchi and Nomura from 2009, stating that maximal abelian covers of regular graphs have no eigenvalues. I will also mention relations between these spectral atoms and those of universal covers, which are a lot easier to analyze as there are much fewer such covers (a single one in case of a regular graph of a fixed degree, in contrast to a whole bunch of covers in case of maximal abelian ones). Based on joint work with Wenbo Li, Michael Magee and Joe Thomas.

The Dry Ten Martini Problem for Sturmian graphs

Gilad Sofer Technion - Israel Institute of Technology Israel Co-Author(s):    Ram Band

Abstract: Are all gaps there? - a question originally posed by Mark Kac for the Almost-Mathieu operator and later termed the Dry Ten Martini Problem by Simon - asks whether all gap labels predicted by the Gap Labelling Theorem are attained by the integrated density of states. For the Almost-Mathieu operator, this problem has been resolved in almost all regimes, and for Sturmian Hamiltonians it was recently answered affirmatively, with all allowed gaps shown to be open. In this talk, we introduce a geometric analogue of this problem for operators on discrete and metric graphs generated by Sturmian subshifts, where the aperiodicity is encoded in the underlying graph structure rather than in a potential. We study ergodic Jacobi operators on such graphs and compute their gap labels explicitly. In contrast to the classical Sturmian Hamiltonians, we show that not all allowed gaps are open. The missing gaps arise from a geometric mechanism rather than a dynamical one: the local structure of the graph produces flat bands, leading to unattained gap labels. In addition, we observe that spectral bands may touch at special energies, due to a degeneracy in the effective dynamics. Based on joint work with Ram Band.

Measure estimates coming from periodic approximations

Lior Tenenbaum Bar-Ilan university Israel Co-Author(s):

Abstract: In studying spectra of complex operators, it is often helpful to work with simpler approximations. In this talk, I will describe a method for estimating the measure of a spectrum that is approximated, in the Hausdorff sense, by spectra of periodic operators. The idea is to consider suitable ``fattenings`` of these spectra and show that their measures converge to that of the limiting spectrum. I will then illustrate this method with an application to aperiodic Schr\odinger operators, whose underlying structures can be approximated by periodic ones.

Exponential mixing for the randomly forced NLS equation

Shengquan Xiang Peking University Peoples Rep of China Co-Author(s):    Yuxuan Chen, Shengquan Xiang, Zhifei Zhang, Jia-Cheng Zhao

Abstract: We explorer exponential mixing of the invariant mesure for randomly forced nonlinear Schroedinger equation, with damping and random noise localized in space. This study emphasizes the crucial role of dispersive nonlinear smoothing and control properties.