"Transport exponents for initial states with large support"

Speaker: 

Vitalii Gerbuz

Institution: 

Rice University

Time: 

Thursday, May 26, 2016 - 2:00pm

One of the classical questions about the evolution of a one
dimensional quantum system is the asymptotic rate of propagation of
the wave packet. It is usually captured through the notion of
transport exponents. Several methods were developed to estimate these
quantities in various models. However many authors only treated the
case of a state initially localized at a single site (in the discrete
setting). We show that some of these results can be extended to a
broad class of initial states with compact or even infinite support,
and explain what are the methods and obstacles to further
generalizations.

The random interchange process on the hypercube

Speaker: 

Roman Kotecky

Institution: 

University of Warwick

Time: 

Tuesday, March 8, 2016 - 3:00pm to 4:00pm

Host: 

Abel Klein

Location: 

RH 340P

We study random permutations of the vertices of a hypercube  given by products of (uniform, independent) random transpositions on edges.  We establish the existence of a phase transition accompanied by emergence of cycles of diverging lengths. The problem is motivated by phase transitions in quantum spin models. (Joint work with Piotr Miłoś and Daniel Ueltschi.)

singular continuous spectrum for singular potentials

Speaker: 

Fan Yang

Institution: 

Ocean Univeristy, visiting UCI

Time: 

Thursday, February 4, 2016 - 2:00pm

Location: 

RH 340P

For singular operators of the form (H_{\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+ \frac{g(\theta+n\alpha)}{f(\theta+n\alpha)} u_n, we prove such operators have purely singular spectrum on the set {E: \delta{(\alpha,\theta)}>L(E)\}, where f and g are both analytic functions.

Singular continuous spectrum

Speaker: 

Fan Yang

Institution: 

Ocean University, visiting UCI

Time: 

Thursday, January 14, 2016 - 2:00pm

Location: 

rh 340p

For singular operators of the form (H_{\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+
\frac{g(\theta+n\alpha)}{f(\theta+n\alpha)} u_n, we prove such operators
have purely singular continuous spectrum on the set
{E: \delta{(\alpha,\theta)}>L(E)\}, where f and g are both analytic functions
on T.

 

 

 
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Singular continuous spectrum for singular potentials

Speaker: 

Fan Yang

Institution: 

Ocean University

Time: 

Thursday, January 7, 2016 - 2:00pm

Location: 

RH 340P

We prove that Schrodinger operators with meromorphic potentials have purely singular continuous spectrum on the set {E: \delta{(\alpha,\theta)}>L(E)\} where \alpha is the frequency, \theta is the phase, delta is an explicit function, and L is the Lyapunov exponent. This extends a result of Jitomirskaya and Liu for the Maryland model to the general class of meromorphic potentials.

Full measure reducibility and localization for quasiperiodic Jacobi operators.

Speaker: 

Rui Han

Institution: 

UCI

Time: 

Thursday, December 10, 2015 - 2:00pm

Location: 

RH 340 P

In this talk we show that if the normalized cocycle related to a quasiperiodic Jacobi operator H_{v,c} (where v being the potential and c being the off-diagonal term) is reducible to a constant rotation for almost all energies with respect to the density of states measure, then its dual model has either purely point spectrum for almost all phase or purely absolutely continuous spectrum for almost all phase, depending on the winding number of c. As a corollary, we obtain the complete phase-transition of extended Harper's model in the positive Lyapunov exponent region.

Reductions of discrete nonlinear wave equations

Speaker: 

Chris Ormerod

Institution: 

Caltech

Time: 

Monday, November 23, 2015 - 2:00pm

Location: 

NSII 1201

 

The exact solutions of the Korteweg-de Vries (KdV) equation obtained by travelling wave and similarity reductions may be expressed in terms of elliptic functions and Painleve transcendents respectively. Discrete versions of the KdV equation may be obtained from chains of commuting Backlund transformations of the KdV equation. These systems are considered integrable in their own right. This introductory talk will demonstrate how solutions obtained as reductions of the discrete KdV equation give us discrete analogues of elliptic equations and discrete Painleve equations, mimicking the case for the KdV equation.

 

Many-body localization and mobility edges (joint Mathematical Physics and Condensed Matter Physics seminar)

Speaker: 

François Huveneers

Institution: 

Université Paris-Dauphine

Time: 

Monday, November 9, 2015 - 1:00pm

Host: 

Abel Klein

Location: 

RH 188

Many-body localization (MBL) generalizes Anderson localization to interacting many-body systems. MBL challenges many fundamental notions of statistical physics. In this talk I will introduce what MBL is from a math-phys perspective. Then I will discuss some striking difference between Anderson localization and MBL, in particular the existence/absence of a mobility edge, i.e. a critical energy (or energy density for many-bod physics) that separates localized states from extended states. (Joint work with W. De Roeck, M. Müller, M. Schiulaz.)

Second phase transition line of the almost Mathieu operator.

Speaker: 

Qi Zhou

Institution: 

Nanjing University

Time: 

Friday, October 16, 2015 - 2:00pm

Location: 

RH 340 P

 

  For the second  phase transition line $\lambda=e^{\beta}$ of  the almost Mathieu operator, we prove that for dense $\alpha$, the operator has purely singular continuous spectrum for every phases, and  for dense $\alpha$, the operator has pure point spectrum for almost every phases. This is joint work with Artur Avila and Svetlana Jitomirskaya. 

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