Positivity of the Lyapunov exponent for potentials generated by hyperbolic transformations

Speaker: 

Zhenghe Zhang

Institution: 

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

Speaker: 

Milivoje Lukic

Institution: 

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.

Approximating the Ground State Eigenvalue via the Landscape Potential

Speaker: 

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

Speaker: 

Wencai Liu

Institution: 

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 $d\geq 2$. 

Entanglement Entropy Bounds in the Higher Spin XXZ Chain

Speaker: 

C. Fischbacher

Institution: 

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-$J$ 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

Speaker: 

Oluwadara Ogunkoya

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

Speaker: 

Lingrui Ge

Institution: 

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.
 

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