# The table and the chair: Spectral approximations beyond dimension one

Siegfried Beckus

## Institution:

University of Potsdam

## Time:

Thursday, December 9, 2021 - 10:00am to 11:00am

## Location:

https://uci.zoom.us/j/97664033581

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 dynamicallydened 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.

# Relations between discrete and continuous spectra of differential operators

Oleg Safronov

UNCC

## Time:

Thursday, May 20, 2021 - 10:00am to 11:00am

## Location:

zoom https://uci.zoom.us/j/93076750122?pwd=Y3pLdndoQTBuNUhxQUxFMkQ2QnRFQT09

We will discuss relations between different parts of spectra of differential operators. In particular, we will see that negative and positive spectra of Schroedinger operators are related to each other. However, there is a stipulation: one needs to consider two operators one of which is obtained from the other by flipping the sign of the potential at each point x. If one knows only that the negative spectra of the two operators are discrete, then their positive spectra do not have gaps. If one knows more about the rate of accumulation of the discrete negative eigenvalues to zero, then one can say more about the absolutely continuous component of the positive spectrum.

# On the spectrum of the Kronig-Penney model in a constant electric field

Simon Larson

Caltech

## Time:

Thursday, May 6, 2021 - 10:00am to 11:00am

## Location:

zoom https://uci.zoom.us/j/93076750122?pwd=Y3pLdndoQTBuNUhxQUxFMkQ2QnRFQT09

We are interested in the nature of the spectrum of the one-dimensional Schr\"odinger operator
$$- \frac{d^2}{dx^2}-Fx + \sum_{n \in \mathbb{Z}}g_n \delta(x-n)$$
with $F>0$ and two different choices of the coupling constants $\{g_n\}_{n\in \mathbb{Z}}$. In the first model $g_n \equiv \lambda$ and we prove that if $F\in \pi^2 \mathbb{Q}$ then the spectrum is $\mathbb{R}$ and is furthermore absolutely continuous away from an explicit discrete set of points. In the second model $g_n$ are independent random variables with mean zero and variance $\lambda^2$. Under certain assumptions on the distribution of these random variables we prove that almost surely the spectrum is dense pure point if $F < \lambda^2/2$ and purely singular continuous if $F> \lambda^2/2$. Based on joint work with Rupert Frank.

# Spectral and dynamical properties of aperiodic quantum walks

Jake Fillman

Texas State

## Time:

Thursday, May 27, 2021 - 10:00am to 11:00am

## Location:

zoom https://uci.zoom.us/j/93076750122?pwd=Y3pLdndoQTBuNUhxQUxFMkQ2QnRFQT09

Quantum walks are quantum mechanical analogues of classical random walks. We will discuss the case of one-dimensional walks in which the quantum coins are modulated by an aperiodic sequence, with an emphasis on almost-periodic models. [Talk based on joint works with Christopher Cedzich, David Damanik, Darren Ong, and Zhenghe Zhang]

# Positivity of the Lyapunov exponent for potentials generated by hyperbolic transformations

Zhenghe Zhang

UCR

## Time:

Thursday, April 29, 2021 - 10:00am to 11:00am

## Location:

zoom https://uci.zoom.us/j/93076750122?pwd=Y3pLdndoQTBuNUhxQUxFMkQ2QnRFQT09

In this talk, I will introduce a recent work in showing positivity of the Lyapunov exponent for Schr\"odinger operators with potentials generated by hyperbolic dynamics. Specifically, we showed that if the base dynamics is a subshift of finite type with an ergodic measure admitting a local product structure and if it has a fixed point, then for all nonconstant H\"older continuous potentials, the set of energies with zero Lyapunov exponent is a discrete set. If the potentials are locally constant or globally fiber bunched, then the set of zero Lyapunov exponent is finite. We also showed that for generic such potentials, we have full positivity in the general case and uniform postivity in the special cases. Such hyperbolic dynamics include expanding maps such as the doubling map on the unit circle, or Anosov diffeomorphism such as the Arnold's Cat map on 2-dimensional torus. It also can be applied to Markov chains whose special cases include the i.i.d. random variable. This is a joint with A. Avila and D. Damanik.

# Reflectionless canonical systems: almost periodicity and character-automorphic Fourier transforms

Milivoje Lukic

Rice University

## Time:

Thursday, April 22, 2021 - 10:00am to 11:00am

## Location:

zoom https://uci.zoom.us/j/93076750122?pwd=Y3pLdndoQTBuNUhxQUxFMkQ2QnRFQT09

This talk describes joint work with Roman Bessonov and Peter
Yuditskii. In the spectral theory of self-adjoint and unitary
operators in one dimension (such as Schrodinger, Dirac, and Jacobi
operators), a half-line operator is encoded by a Weyl function; for
whole-line operators, the reflectionless property is a
pseudocontinuation relation between the two half-line Weyl functions.
We develop the theory of reflectionless canonical systems with an
arbitrary Dirichlet-regular Widom spectrum with the Direct Cauchy
Theorem property. This generalizes, to an infinite gap setting, the
constructions of finite gap quasiperiodic (algebro-geometric)
solutions of stationary integrable hierarchies. Instead of theta
functions on a compact Riemann surface, the construction is based on
reproducing kernels of character-automorphic Hardy spaces in Widom
domains with respect to Martin measure. We also construct unitary
character-automorphic Fourier transforms which generalize the
Paley-Wiener theorem. Finally, we find the correct notion of almost
periodicity which holds in general for canonical system parameters in
Arov gauge, and we prove generically optimal results for almost
periodicity for Potapov-de Branges gauge, and Dirac operators.

# Order amidst Disorder: Hidden 1D equations in 2D and 3D quantum models with arbitrarily strong disorder

Mark Novotny

## Institution:

Mississippi State, Physics

## Time:

Thursday, April 8, 2021 - 10:00am to 11:00am

## Location:

https://uci.zoom.us/j/93076750122?pwd=Y3pLdndoQTBuNUhxQUxFMkQ2QnRFQT09

# Approximating the Ground State Eigenvalue via the Landscape Potential

Shiwen Zhang

## Institution:

University of Minnesota

## Time:

Thursday, April 15, 2021 - 10:00am to 11:00am

## Location:

zoom https://uci.zoom.us/j/93076750122?pwd=Y3pLdndoQTBuNUhxQUxFMkQ2QnRFQT09

In this talk, we study the ground state energy of a Schrodinger operator and its relation to the landscape potential. For the 1-d Bernoulli Anderson model, we show that the ratio of the ground state energy and the minimum of the landscape potential approaches pi^2/8 as the domain size approaches infinity. We then discuss some numerical stimulations and conjectures for excited states and for other random potentials. The talk is based on joint work with I. Chenn and W. Wang.

# Irreducibility of the Fermi variety for discrete periodic Schr\"odinger operators

Wencai Liu

Texas A&M

## Time:

Thursday, February 11, 2021 - 10:00am to 11:00am

## Location:

zoom https://uci.zoom.us/j/93076750122?pwd=Y3pLdndoQTBuNUhxQUxFMkQ2QnRFQT09

Let $H_0$ be a discrete periodic  Schr\"odinger operator on $\Z^d$:

$$H_0=-\Delta+V,$$ where $\Delta$ is the discrete Laplacian and $V:\Z^d\to \R$ is periodic.    We prove that  for any $d\geq3$,    the Fermi variety at every energy level  is irreducible  (modulo periodicity).  For $d=2$,    we prove that the Fermi variety at every energy level except for the average of  the potential    is irreducible  (modulo periodicity) and  the Fermi variety at the average of  the potential has at most two irreducible components  (modulo periodicity).

This is sharp since for  $d=2$ and a constant potential  $V$,

the Fermi variety at  $V$-level  has exactly  two irreducible components (modulo periodicity).

In particular,  we show that  the Bloch variety  is irreducible

(modulo periodicity)  for any $d\geq 2$.

# Something Cool

Rui Han

LSU

## Time:

Thursday, June 24, 2021 - 10:00am to 11:00am

## Location:

zoom https://uci.zoom.us/j/93076750122?pwd=Y3pLdndoQTBuNUhxQUxFMkQ2QnRFQT09