On the relation between strong ballistic transport and exponential dynamical localization

Speaker: 

Ilya Kachkovskiy

Institution: 

MSU

Time: 

Thursday, March 5, 2020 - 2:00am to 3:00am

Host: 

Location: 

RH 340

Abstract: We establish strong ballistic transport for a family of discrete quasiperiodic Schrodinger operators as a consequence of exponential dynamical localization for the dual family. The latter has been, essentially, shown by Jitomirskaya and Kruger in the one-frequency setting and by Ge--You--Zhou in the multi-frequency case. In both regimes, we obtain strong convergence of $\frac{1}{T}X(T)$ to the asymptotic velocity operator $Q$, which improves recent perturbative results by Zhao and provides the strongest known form of ballistic motion. In the one-frequency setting, this approach allows to treat Diophantine frequencies non-perturbatively and also consider the weakly Liouville case.

The proof is based on the duality method. Originally, localization for the dual model allows to obtain ballistic transport in expectation. Combined with dynamical localization bounds, the improved convergence allows to replace ``in expectation’’ by ``almost surely’'.

Logarithmic lower bounds for the entanglement entropy of droplet states for the XXZ model on the ring

Speaker: 

Ruth Schulte

Institution: 

Ludwig-Maximilians-Universitat Munich

Time: 

Thursday, February 27, 2020 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

We study the free XXZ quantum spin model defined on a ring of size L and show that the bipartite entanglement entropy of eigenstates belonging to the first energy band above the vacuum ground state satisfy a logarithmically corrected area law. 

 

Speculative and hedging interaction model in oil and U.S. dollar markets—Long-term investor dynamics and phases

Speaker: 

Michael Campbell

Institution: 

Eureka (SAP)

Time: 

Thursday, January 9, 2020 - 2:00pm to 3:00pm

Location: 

RH 340 P

We develop the rational dynamics for the long-term investor among boundedly rational speculators in the Carfì–Musolino speculative and hedging model. Numerical evidence is given that indicates there are various phases determined by the degree of nonrational behavior of speculators. The dynamics are shown to be influenced by speculator “noise”. This model has two types of operators: a real economic subject (Air, a long-term trader) and one or more investment banks (Bank, short-term speculators). It also has two markets: oil spot market and U.S. dollar futures. Bank agents react to Air and equilibrate much more quickly than Air, thus we consider rational, best-local-response dynamics for Air based on averaged values of equilibrated Bank variables. The averaged Bank variables are effectively parameters for Air dynamics that depend on deviations-from-rationality (temperature) and Air investment (external field). At zero field, below a critical temperature, there is a phase transition in the speculator system which creates two equilibriums for bank variables, hence in this regime the parameters for the dynamics of the long-term investor Air can undergo a rapid change, which is exactly what happens in the study of quenched dynamics for physical systems. It is also shown that large changes in strategy by the long-term Air investor are always preceded by diverging spatial volatility of Bank speculators. The phases resemble those for unemployment in the “Mark 0” macroeconomic model.

 

Eigensystem multiscale analysis for the Anderson model via the Wegner estimate

Speaker: 

Abel Klein

Institution: 

UCI

Time: 

Thursday, October 3, 2019 - 2:00pm

Host: 

Location: 

RH 340P

We present a new approach to the  eigensystem multiscale analysis (EMSA) for the Anderson model that relies on the Wegner estimate.  The EMSA  treats all energies  of the finite volume operator in an energy interval at the same time,  simultaneously establishing localization of  all eigenfunctions with eigenvalues in the energy interval  with high probability.  It implies all the usual manifestations of localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization)   for the Anderson model. The new method removes the restrictive level spacing hypothesis used in the previous versions of the EMSA, allowing  for single site probability distributions that are  H\"older continuous of order  $\alpha \in (0,1]$. (Joint work with Alex Elgart.)

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

Speaker: 

Xin Zhao

Institution: 

Nanjing Universuty

Time: 

Thursday, October 10, 2019 - 2:00pm to 3:00pm

Location: 

rh 340p

In this talk, we first consider quasi-periodic Schr\"odinger operators with finitely differentiable potentials. If the potential is analytic, there are numerous results. But not every result holds if one replaces the analyticity with a smoothness condition. We will give some positive results in this aspect,  generalizing some interesting results in the analytic case to the finitely smooth case. This includes the global reducibility results, generalized Chamber's formula and their applications to the study of continuity of the spectra. Finally we will give a recent result on the continuity of spectral measure of multi frequency quasi-periodic Schr\"odinger operators with small analytic quasi-periodic potentials.

Landscape theory for tight-binding Hamiltonians 

Speaker: 

Shiwen Zhang

Institution: 

U Minnesota

Time: 

Friday, November 15, 2019 - 1:00pm to 2:00pm

Location: 

rh 340p

In 2012, Filoche and Mayboroda introduced the concept of the landscape function u, for an elliptic operator L, which solves the inhomogeneous equation Lu=1. This landscape function has remarkable power to predict the shape and location of localized low energy eigenfunction. These ideas led to beautiful results in mathematics, as well as theoretical and experimental physics. In this talk, we first briefly review these results of landscape theory for differential operators on R^d. We will then discuss some recent progress of extending landscape theory to tight-binding Hamiltonians on discrete lattice Z^d. In particular, we show that the effective potential 1/u creates barrier for appropriate exponential decay eigenfunctions of Agmon type for some discrete Schrodinger operators. We also show that the minimum of 1/u leads to a new counting function, which gives non-asymptotic estimates on the integrated density of states of the Schrodinger operators. This talk contains joint work in progress with S. Mayboroda and some numerical experiments with W. Wang.

The Christ-Kiselev's multi-linear operator technique and its applications to Schrodinger operators

Speaker: 

Wencai Liu

Institution: 

Texas A&M

Time: 

Thursday, November 14, 2019 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

We established an axiomatic version of Christ-Kiselev's multi-linear operator techniques.
As applications,  several spectral results of perturbed periodic Schrodinger operators are obtained, including WKB solutions, sharp transitions of preservation of absolutely continuous spectra, criteria of absence of singular spectra and sharp bounds of Hausdorff dimensions of singular spectra.

Ergodic Schrodinger operators in the infinite-measure setting

Speaker: 

Jake Fillman

Institution: 

Texas State University

Time: 

Thursday, October 24, 2019 - 2:00pm to 3:00pm

Location: 

RH 340P

We develop the basic spectral theory of ergodic Schrodinger operators when the underlying dynamics are given by a conservative
ergodic transformation of a \sigma-finite measure space. Some fundamental results, such as the Ishii--Pastur theorem carry over to the
infinite-measure setting. We also discuss some examples in which straightforward analogs of results from the probability-measure case do not hold. We will discuss some examples and some interesting open problems.

The talk is based on a joint work with M. Boshernitzan, D. Damanik, and M. Lukic.

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