# Bose-Einstein condensation in one-dimensional noninteracting Bose gases in the presence of soft Poissonian obstacles

## Speaker:

Maximilian Pechmann

## Institution:

University of Tennessee, Knoxville

## Time:

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

We study Bose--Einstein condensation (BEC) in one-dimensional noninteracting Bose gases in Poisson random potentials on $\mathbb R$ with single-site potentials that are nonnegative, compactly supported, and bounded measurable functions in the grand-canonical ensemble at positive temperatures and in the thermodynamic limit. For particle densities larger than a critical one, we prove the following: With arbitrarily high probability when choosing the fixed strength of the random potential sufficiently large, BEC where only the ground state is macroscopically occupied occurs. If the strength of the Poisson random potential converges to infinity in a certain sense but arbitrarily slowly, then this kind of BEC occurs in probability and in the $r$th mean, $r \ge 1$. Furthermore, in Poisson random potentials of any fixed strength an arbitrarily high probability for type-I g-BEC is also obtained by allowing sufficiently many one-particle states to be macroscopically occupied.

# Smooth quasiperiodic SL(2,\R)-cocycles (II)-Sharp transition space for the continuity of the Lyapunov exponent.

Lingrui Ge

UCI

## Time:

Thursday, January 28, 2021 - 10:00am to 11:00am

## Location:

https://uci.zoom.us/j/95453091338?pwd=L1hXRTRremw4YjFQY3I3NHZGNUdKZz09

We construct discontinuous points of the Lyapunov exponent of quasiperiodic Shr\"odinger cocycles in Gevrey space $G^{s}$ with $s>2$. In contrast, the Lyapunov exponent has been proved to be continuous in $G^{s}$ with $s<2$ by Klein and Cheng-Ge-You-Zhou. This shows that $G^2$ is the transition space for the continuity of the Lyapunov exponent.

# Smooth quasiperiodic SL(2,\R)-cocycles (I)-Global rigidity results for rotations reducibility and Last's intersection spectrum conjecture.

Lingrui Ge

UCI

## Time:

Thursday, January 21, 2021 - 10:00am to 11:00am

## Location:

https://uci.zoom.us/j/97333959480?pwd=VExLT1d2Q0F6SkovT3hmbGhBZ05HUT09

For quasiperiodic Schr\"odinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schr\"odinger cocycle is either rotations reducible or has positive Lyapunov exponent for all irrational frequency and almost every energy by Avila-Fayad-Krikorian. From spectral theory side, the Schr\"odinger conjecture"  has been verified by Avila-Fayad-Krikorian and the Last's intersection spectrum conjecture" has been proved by Jitomirskaya-Marx. The proofs of above results crucially depend on the analyticity of the potentials. People are curious about if the analyticity is essential for those problems, some open problems in this aspect were raised by  Fayad-Krikorian and Jitomirskaya-Marx. In this paper, we prove the above mentioned results for ultra-differentiable potentials.

# On the abominable properties of the Almost Mathieu operator with Liouville frequencies

Mira Shamis

## Institution:

Queen Mary, london

## Time:

Thursday, January 14, 2021 - 12:00pm to 1:00pm

## Location:

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

We show that, for sufficiently well approximable frequencies, several spectral characteristics of the Almost Mathieu operator can be as poor as at all possible in the class of all discrete Schroedinger operators. For example, the modulus of continuity of the integrated density of states may be no better than logarithmic. Other characteristics to be discussed are homogeneity, the Parreau-Widom property, and (for the critical AMO) the Hausdorff content of the spectrum. Based on joint work with A. Avila, Y. Last, and Q. Zhou

# THE UMPTEEN OPERATOR

Sasha Sodin

## Institution:

Queen Mary, london

## Time:

Thursday, November 12, 2020 - 12:30pm to 1:30pm

## Location:

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

Abstract:

It was found in the 1990s that special linear maps playing a role in the representation theory of the symmetric group share common features with random matrices. We construct a representation-theoretic operator which shares some properties with the Anderson model (or, perhaps, with magnetic random Schroedinger operators), and show that indeed it boasts Lifshitz tails. The proof relies on a close connection between the operator and the infinite board version of the fifteen puzzle.
No background in the representation theory of the symmetric group will be assumed. Based on joint work with Ohad Feldheim.

# Continuity of spectra and spectral measure of quasi-periodic Schr\"odinger operators, 2

Xin Zhao

## Institution:

Nanjing University

## Time:

Thursday, October 31, 2019 - 2:00pm

RH 340

# Speculative and hedging interaction statistical mechanics model in oil and U.S. dollar markets

Michael Campbel

Eureka AI

## Time:

Thursday, January 16, 2020 - 2:00pm

Rh 340

# The spectrum of the critical almost Mathieu operator in the rational case

Igor Krasovsky

Imperial College

## Time:

Friday, February 28, 2020 - 2:00pm

RH 340

# Continuity of spectra and spectral measure of quasi-periodic Schr\"odinger operators, III

Xin Zhao

Nanjing

## Time:

Saturday, December 19, 2020 - 2:00pm

RH 340

# Continuity of spectra and spectral measure of quasi-periodic Schr\"odinger operators, II

Xin Zhao

Nanjing

## Time:

Saturday, October 31, 2020 - 2:00pm

RH 340