Schedule for: 22w5163 - Almost-Periodic Spectral Problems

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 3rd floor of the Sally Borden.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will describe a theory of regularity for one-dimensional continuum Schr\"odinger operators and Dirac operators whose potentials are bounded in an appropriate local norm. The theory is based on the Martin compactification of the complement of the essential spectrum. It gives universal inequalities for the thickness of the spectrum and exponential growth rate of Dirichlet solutions; on the other side of the inequalities are potential theoretic notions such as Martin functions and new renormalized Robin constants found in the asymptotics at infinity. I will also discuss applications to decaying and ergodic potentials, and interesting differences between the Schrodinger and Dirac settings. The talk is based on joint work with Benjamin Eichinger and Ethan Gwaltney.

We introduce a unitary almost-Mathieu operator, which is a one-dimensional quasi-periodic quantum walk obtained from an anisotropic two-dimensional quantum walk in a uniform magnetic field. We will discuss background information, the origins of the model, its interesting spectral features, and some ideas needed in proofs of the main results. [Joint work with Christopher Cedzich, Darren C. Ong, and Zhenghe Zhang]

In this talk we introduce a Nash-Moser type iteration approach to establish power-law localization for some discrete almost-periodic operators with polynomial long-range hopping. We also provide a quantitative lower bound on the regularity of the long-range hopping.

For 1-d quantum harmonic oscillator perturbed by a time quasi-periodic polynomial of (x,-i\partial_x) of degree 2, we consider its almost reducibility. As an application, we show different behaviors of the solutions in Sobolev spaces. This is based on several joint works with Z. Liang, J. Luo and Q. Zhou.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will discuss new results on reducibility of quasi-periodic cocycles taking their values in symplectic groups that might be interesting in the study of the spectrum of Schrödinger operators on strips. I will in particular focus on the following non perturbative almost sure dichotomy result: for natural one parameter families of such cocycles, a.s with respect to the parameter, either the maximal Lyapunov exponent is positive or the cocycle is almost reducible to some model cocycle. The techniques of the proof combines Kotani theory, renormalization techniques and KAM theory. This is a joint work with Artur Avila and Yi Pan.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

This talk will be on the spectral properties of elastic beam Hamiltonian defined on periodic hexagonal lattices. These continua are constructed out of Euler-Bernoulli beams, each governed by a scalar valued fourth-order Schrödinger operator equipped with a real periodic symmetric potential. Unlike the second-order Schrödinger operator commonly applied in quantum graph literature, here the self-adjoint vertex conditions encode geometry of the graph by their dependence on angles at which edges are met. I will consider spectral properties of this Hamiltonian on a special equal-angle lattice, known as graphene or honeycomb lattice. This talk is based on a recent joint work with Mahmood Ettehad (University of Minnesota), https://arxiv.org/pdf/2110.05466.pdf.

Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. I will present a very simple operator whose spectral properties are thought to determine which angles are magical. It comes from a 2019 PR Letter by Tarnopolsky--Kruchkov--Vishwanath. The mathematics behind this is an elementary blend of representation theory (of the Heisenberg group in characteristic three), Jacobi theta functions and spectral instability of non-selfadjoint operators (involving Hörmander's bracket condition in a very simple setting). Recent mathematical progress also includes the proof of the existence of magic angles. The results will be illustrated by colourful numerics which suggest many open problems. Connections to the world of quasi-periodic operators will be outlined.

Sato proposed a unified constructive way of treating a certain class of non-linear PDEs in 1980. Later in 1985 his idea was realized as a flow on a Grassmann manifold $Gr^{(\nu)}$ consisting of all closed subspaces $W$ of $L^{2}\left( \left\vert z\right\vert =r\right)$ satisfying $z^{\nu}W\subset W$. If $\nu=2$, this flow generates the KdV equation. For $\nu=2$ the speaker extended this framework to $L^{2}\left( C\right)$ on an unbounded curve $C$, and obtained solutions to the KdV equation with large class of initial data including ergodic ones. In this talk we remark that this procedure is available also for the Boussinesq equation and the non-linear Schr\"{o}dinger equation, and present several open questions.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 3rd floor of the Sally Borden.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Let $\Delta+V$ be the discrete Schr\"odinger operator, where $\Delta$ is the discrete Laplacian on $\Z^d$ and the potential $V:\Z^d\to \R$ is $\Gamma$-periodic. We prove three rigidity theorems for discrete periodic Schr\"odinger operators in any dimension $d\geq 3$: $$\begin{enumerate} \item if $V$ and $Y$ are Fermi isospectral (that is, at some energy level, Fermi varieties of the $\Gamma$-periodic potential $V$ and the $\Gamma$-periodic potential $Y$ are the same), and $Y $ is a separable function, then $V$ is separable as well; \item if potentials $V$ and $Y$ are Fermi isospectral and both $V=\bigoplus_{j=1}^rV_j$ and $Y=\bigoplus_{j=1}^r Y_j$ are separable functions, then, up to a constant, lower dimensional decompositions $V_j$ and $Y_j$ are Floquet isospectral, $j=1,2,\cdots,r$; \item if a potential $V$ and the zero potential are Fermi isospectral, then $V$ is zero. \end{enumerate}$$

We shall discuss the quantum dynamics associated with ergodic Schroedinger operators with singular continuous spectrum. Upper bounds on the transport moments have been obtained for several classes of one-dimensional operators, particularly, by Damanik--Tcheremchantsev, Jitomirskaya--Liu, Jitomirskaya--Powell. We shall present a new method which allows to recover most of the previous results and also to obtain new results in one and higher dimensions. The input required to apply the method is a large-deviation estimate on the Green function at a single energy. Based on joint work with S. Sodin.

The purpose of this talk is to describe a series of works on mixed random-quasiperiodic linear cocycles, establishing results of Furstenberg's theory (e.g. a criterion for the positivity of the top Lyapunov exponent) and large deviations estimates. One of our main motivations in the study of such systems is the stability of Lyapunov exponents of quasiperiodic cocycles under random noise, which addresses a question posed by J. You. Joint work with A. Cai (PUC-Rio) and P. Duarte (University of Lisbon).

The table and the chair tiling are two aperiodic tilings of the plane that are typical examples of two-dimensional quasicrystals. One way to treat such systems in dimension one, is to approximate these systems by suitable (periodic) approximations. Based on this, we raise the following questions: Is there a general method to approximate spectral properties of a given operator by the underlying geometry or dynamics? If so, can we control the approximations and which spectral properties are preserved? During the talk, we provide a short overview over such results with a special focus on dynamically defined operator families. We will see as how to apply those results explicitly and what they tell us about the table and the chair tiling. These results are joint works with Ram Band, Jean Bellissard, Horia Cornean, Giusseppe De Nittis, Felix Pogorzelski, Alberto Takase and Lior Tenenbaum.

Around 2007, Germinet and Müller proposed that the Multiscale Analysis method used to prove localization for random operators could shed some light on the localization properties of deterministic Delone operators, that is, Schrödinger operators where the potential is defined on a Delone set and is therefore not necessarily defined on a lattice. This initiated a research program where, as a stepping stone, we set out to study the ergodic and spectral properties of random operators where the random potential is defined on a Delone set. Understanding the particular case of Bernoulli disorder allowed us to make the crucial link and to give an answer to the original question. In this talk we will review the outcome of this program, and discuss results on ergodic and spectral properties of (random) Delone operators.

The purpose of this talk is to describe recent and ongoing work regarding Lyapunov exponents for analytic quasiperiodic operators with finite-valued backgrounds. We prove that, for sufficiently large coupling constant, the Lyapunov exponent is positive with a uniform minoration, based on a method that goes back to Bourgain. Moreover, we show that the coupling constant can be taken independent of the background in the periodic case.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 3rd floor of the Sally Borden.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Over the last 15 years, many developments in the spectral theory of multidimensional almost-periodic operators were obtained using a well-adapted gauge transform. In particular, we could mention asymptotics for the integrated and local density of states, or resolution of Bethe--Sommerfeld type problems. The gauge transforms were in general ad hoc and precise definitions needed to change from paper to paper. In this talk, I will present a general unified method and give some examples of spectral results that were obtained through this general method, for instance in the case of almost periodic Dirac operators.

The question of high energy asymptotics for the kernel of the spectral projector of the Laplacian in the context of compact manifolds is one of the most well studied areas of spectral theory since the early 1900s. In this talk, we discuss the analogous question for Schrodinger operators on the real line: What are the asymptotics for the spectral projector of a Schrodinger operator on \mathbb{R}? By combining the classical wave method, originally introduced by Levitan in the 1950s, with the periodic gauge transform technique, we are able to show that when the potential is bounded with all derivatives this kernel, known as the local density of states, has a full asymptotic expansion in powers of the spectral parameter. This proves a conjecture of Parnovski--Shterenberg in the one dimensional case.

We consider Schr\"{o}dinger operators $H=-\Delta +V(x)$ in $\mathbb R^d$, $d\geq 2$, with quasi-periodic potentials $V(x)$. We prove that the absolutely continuous spectrum of a generic H contains a semi-axis $[\lambda_*,\infty)$. We also construct a family of eigenfunctions of the absolutely continuous spectrum; these eigenfunctions are small perturbations of the exponentials. The proof is based on the multi-scale analysis in the momentum space.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 3rd floor of the Sally Borden.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In the past decades, rich results on dynamical and spectral properties of analytic quasi-periodic Schrodinger operators (QPSO), like positivity and regularity of Lyapunov exponents (LE), almost reducibility and almost localization, Cantor spectrum etc., have been obtained. The methods among these results depend heavily on the regularity condition. On the other hand, recently we obtained the counterpart of the above results for C^2 large-coupling QPSO imposed with a so called cos-type geometry condition on the potentials. Among them, some property is even not true for general analytic QPSO. For example, we will show that LE is 1/2 -Holder continuous and absolutely continuous with respect to the energy for C^2 cos-type potentials with a large coupling, while the regularity of LE for general analytic potentials can be less than delta-Holder continuous for any small delta > 0. In this talk, we will discuss the roles played by regularity condition and geometry condition in QSPO.

We prove the multiplicative Jensen's formula for one-frequency Schr\"odinger cocycles. Our formula gives a quantitative version of Avila's global theory, including new equivalent definitions of subcritical, critical, supercritical regimes and quantitative characterization of the acceleration. We also drive several spectral and physical applications. This is based on a joint work with Jitomirskaya, You and Zhou.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider quasiperiodic operators on $\Z^d$ with unbounded monotone sampling functions ("Maryland-type”), and construct the Rayleigh—Schrodinger formal perturbation series for the eigenvalues and eigenvectors. We discuss the combinatorial structure and cancellations in such series and sufficient conditions for their convergence. This allows to establish Anderson localization for several classes of mononone quasiperiodic operators by constructing explicit converging expansions for eigenvalues and eigenvectors. If time permits, we will discuss cases where the requirement of strict monotonicity or unboundedness can be relaxed. The talk is based on the joint work with S. Krymskii, L. Parnovskii, and R. Shterenberg, both published and in progress.

Quantum spin models are a paradigm for the study of many-body effects. In this talk, we will discuss some recent results on the XY spin chain in quasi-periodic exterior magnetic field. In particular, we show dynamical localization for a family of anisotropic XY chains for all Diophantine magnetic fluxes, in the sense that the time evolution of local observables satisfies a zero-velocity Lieb-Robinson bound.

We review results on nonlinear Anderson localization. This talk is based on the joint works with J. Bourgain, and more recently with W. Liu.

In this talk, I will introduce the existence and uniqueness of spatially quasi-periodic solutions to the generalized KdV equation on the real line. By applying a combinatorial analysis method, we solve the infinite dimensional ODEs generated by the PDE and obtain the local result. This is based on the joint work with David Damanik (Rice University) and Yong Li (Jilin University).

Quantum walks are quantum analogies of the classic random walks. In this talk, we will discuss some recent results, in particular Anderson localization for all Diophantine frequencies, for quantum walk model arising from 2d magnetic operator.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 3rd floor of the Sally Borden.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will talk about several phase transition problem for (non-self adjoint) quasi-periodic Schrodinger operator, these topics receive wide attention of the physicist recent years. Starting from almost Mathieu operator, i will introduce several physical models, and talk about their phase transition results, including mobility edge, PT symmetry transition and topological phase transition.

In this talk, we prove absolute continuity of the integrated density of states for frequency-independent analytic perturbations of the non-critical almost Mathieu operator under arithmetic conditions on frequency. This is based on a joint work with Lingrui Ge and Svetlana Jitomirskaya.

In this talk we compare the CMV matrices with discrete Schrodinger operators in the quasi-periodic setting and prove the purely absolutely continuous spectrum for extended CMV matrices for every phase. We also establish a criterion to prove the analyticity of the tongue boundaries and apply it to the CMV matrices with near constant almost periodic VC’s. As a consequence, we prove the generic dry version of Cantor spectrum of associated extended CMV matrices.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.