# 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

# Something Cool

Ilya Kachkovskiy

MSU

## Time:

Thursday, February 18, 2021 - 10:00am to 11:00am

## Location:

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

# Entanglement Entropy Bounds in the Higher Spin XXZ Chain

C. Fischbacher

UCI

## Time:

Thursday, March 4, 2021 - 10:00am to 11:00am

## Location:

https://uci.zoom.us/j/93076750122?pwd=Y3pLdndoQTBuNUhxQUxFMkQ2QnRFQT09
This is the second part of a series of two talks. We consider the Heisenberg XXZ spin-$J$&nbsp;chain ($J\in\mathbb{N}/2$) with anisotropy parameter $\Delta$. Assuming that $\Delta>2J$, and introducing threshold energies $E_{K}:=K\left(1-\frac{2J}{\Delta}\right)$, we show that the bipartite entanglement entropy (EE) of states belonging to any spectral subspace with energy less than $E_{K+1}$ satisfy a logarithmically corrected area law with prefactor $(2\lfloor K/J\rfloor-2)$. This generalizes previous results by Beaud and Warzel as well as Abdul-Rahman, Stolz, and CF who covered the spin-$1/2$ case.

# Entanglement Entropy Bounds in the Higher Spin XXZ Chain

## Institution:

University of Alabama at Birmingham

## Time:

Thursday, February 25, 2021 - 10:00am to 11:00am

## Location:

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

We consider the Heisenberg XXZ spin-$J$ chain ($J\in\mathbb{N}/2$) with anisotropy parameter $\Delta$. Assuming that $\Delta>2J$, and introducing threshold energies $E_{K}:=K\left(1-\frac{2J}{\Delta}\right)$, we show that the bipartite entanglement entropy (EE) of states belonging to any spectral subspace with energy less than $E_{K+1}$ satisfy a logarithmically corrected area law with prefactor $(2\lfloor K/J\rfloor-2)$.

This generalizes previous results by Beaud and Warzel as well as Abdul-Rahman, Fischbacher and Stolz, who covered the spin-$1/2$ case.

# Smooth quasiperiodic SL(2,\R)-cocycles (III)-Sketch of the proof of the continuity of the Lyapunov exponent

Lingrui Ge

UCI

## Time:

Thursday, February 4, 2021 - 10:00am to 11:00am

## Location:

Join Zoom Meeting https://uci.zoom.us/j/96960481956

In this talk, we will sketch how to generalize the large deviation theorem to quasiperiodic Gevrey $SL(2,\R)$-cocycles and use it to prove the joint continuity of the Lyapunov exponent.

# 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.