# Anderson Localization for Schrödinger Operators with Monotone Potentials over Circle Diffeomorphisms, 2

Jiranan Kerdboon

## Institution:

Mississippi State

## Time:

Friday, June 3, 2022 - 1:00pm to 2:00pm

## Location:

306

We generalize localization results on 1D quasiperiodic Schrödinger operators with monotone potentials over Diophantine irrational rotations to the results over circle diffeomorphisms with irrational rotation numbers. Our results show that the class of irrational rotation numbers can be extended to weakly Liouville irrat

# Anderson Localization for Schrödinger Operators with Monotone Potentials over Circle Diffeomorphisms

Jiranan Kerdboon

U Mississippi

## Time:

Tuesday, June 7, 2022 - 2:00pm to 3:00pm

## Location:

306

We generalize localization results on 1D quasiperiodic Schrödinger operators with monotone potentials over Diophantine irrational rotations to the results over circle diffeomorphisms with irrational rotation numbers. Our results show that the class of irrational rotation numbers can be extended to weakly Liouville irrational rotation numbers.

# The multispecies zero range process and modified Macdonald polynomials

Olya Mandelshtam

U Waterloo

## Time:

Monday, November 29, 2021 - 2:00pm to 3:00pm

## Location:

RH 510R

Over the last couple of decades, the theory of interacting particle systems has found some unexpected connections to orthogonal polynomials, symmetric functions, and various combinatorial structures. The asymmetric simple exclusion process (ASEP) has played a central role in this connection. Recently, Cantini, de Gier, and Wheeler found that the partition function of the multispecies ASEP on a circle is a specialization of a Macdonald polynomial $P_{\lambda}(X;q,t)$. Macdonald polynomials are a family of symmetric functions that are ubiquitous in algebraic combinatorics and specialize to or generalize many other important special functions. Around the same time, Martin gave a recursive formulation expressing the stationary probabilities of the ASEP on a circle as sums over combinatorial objects known as multiline queues, which are a type of queueing system. Shortly after, with Corteel and Williams we generalized Martin's result to give a new formula for $P_{\lambda}$ via multiline queues.

The modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$ are a version of $P_{\lambda}$ with positive integer coefficients. A natural question was whether there exists a related statistical mechanics model for which some specialization of $\widetilde{H}_{\lambda}$ is equal to its partition function. With Ayyer and Martin, we answer this question in the affirmative with the multispecies totally asymmetric zero-range process (TAZRP), which is a specialization of a more general class of zero range particle processes. We introduce a new combinatorial object in the flavor of the multiline queues, which on one hand, expresses stationary probabilities of the mTAZRP, and on the other hand, gives a new formula for $\widetilde{H}_{\lambda}$. We define an enhanced Markov chain on these objects that lumps to the multispecies TAZRP, and then use this to prove several results about particle densities and correlations in the TAZRP.

# On Vertex Matching Conditions in Elastic Beam Frames

U Minnesota

## Time:

Thursday, May 5, 2022 - 2:00pm to 3:00pm

## Location:

rh 306 plus zoom

Abstract: Modeling elastic frames constructed out of beam elements is of natural interest to engineers working on structural analysis discipline. From a more theoretical angle, this problem may be viewed as an analysis of a differential operator (Hamiltonian) acting on a metric graph in which the question of describing correct matching conditions is of central importance. We start this talk by considering three-dimensional elastic frames constructed out of Euler–Bernoulli beams and describe the notion of rigidity at a joint, i.e., the case in which relative angles of participating beams remain constant throughout the motion. Next, we discuss extension of matching conditions by relaxing the vertex-rigidity assumption and the case in which concentrated mass may exist. This generalization is based on coupling an (elastic) energy functional in terms of field’s discontinuities at a vertex along with purely geometric terms derived out of first principles.

This talk is based on joint works with Gregory Berkolaiko (Texas A&M University) and Soohee Bae (Northeastern University).

# Spectral Properties of Periodic Elastic Beam Lattices

Burak Hatinoglu

UCSB

## Time:

Friday, April 8, 2022 - 1:00pm

## Location:

RH 305

Abstract: 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 firstly consider this Hamiltonian on a special equal-angle lattice, known as graphene or honeycomb lattice. I will also discuss spectral properties for the same operator on lattices in the geometric neighborhood of graphene. This talk is based on a recent joint work with Mahmood Ettehad (University of Minnesota), https://arxiv.org/pdf/2110.05466.pdf

# On the Correspondence Between the Spectrum of Jacobi Operators and Domination of Jacobi Cocycles

Zhenghe Zhang

## Institution:

University of California, Riverside

## Time:

Thursday, March 17, 2022 - 2:00pm to 3:00pm

## Location:

RH 306

In this talk we first introduce a notion of dominated splitting for M(2,C) sequences and show an energy parameter belongs to the spectrum of a Jacobi operator, possibly singular, if and only if the associated Jacobi cocycle does not admit dominated splitting. Then we consider dynamically defined Jacobi operators whose base dynamics is only assumed to be topologically transitive. We show an energy parameter belongs to the spectrum of the operator defined by a base point with a dense orbit if and only if the dynamically defined Jacobi cocycle does not admit dominated splitting. This extends both the original Johnson's theorem for Schrodinger operators and a previous result obtained by Chris Marx for certain dynamically defined Jacobi operators. This is a joint work with Kateryna Alkorn.

# 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]