THE UMPTEEN OPERATOR

Speaker: 

Sasha Sodin

Institution: 

Queen Mary, london

Time: 

Thursday, November 12, 2020 - 11:30am to 12:30pm

Location: 

zoom

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.

Dynamical delocalization for discrete Schrödinger operators

Speaker: 

Simon Becker

Institution: 

Cambridge University

Time: 

Tuesday, March 17, 2020 - 2:00pm

We study discrete magnetic random Schrödinger operators on the square and honeycomb lattice, both with single-site potentials in weak magnetic fields under weak disorder. We show that there is, in the case of the honeycomb lattice, both strong dynamical localization and delocalization close to the conical point. We obtain similar results for the discrete random Schrödinger operator on the Z2-lattice close to the bottom and top of its spectrum. As part of this analysis, we give a rigorous derivation of the quantum hall effect for both models derived from the density of states for which we obtain an asymptotic expansion in the disorder parameter. The expansion implies (leading order in the disorder parameter) universality of the integrated density of states. Finally, we show that on the hexagonal lattice the Dirac cones occur for any rational magnetic flux. We use this observation to study the self-similarity of the Hall conductivity and transport properties of the random operator close to any rational magnetic flux.

Joint work with Rui Han.

 Irreducibility of Fermi variety for periodic Schr\"odinger operators in higher dimensions

Speaker: 

Wencai Liu

Institution: 

TAMU

Time: 

Thursday, March 12, 2020 - 2:00pm to 3:00pm

Location: 

RH 340

This is an ongoing project.  Let $H_0$ be a discrete periodic  Schr\"odinger operator on $\Z^d$:

$$H_0=-\Delta+v_0,$$

where $-\Delta$ is the discrete Laplacian and $v_0$ is periodic in the sense that it is well defined on  $\Z^d/q_1\Z\oplus q_2 \Z\oplus\cdots\oplus q_d\Z$. For $d=2$, we tentatively proved that the Fermi variety $F_{\lambda}(v_0)/\Z^2$ is irreducible except for one value of  $\lambda$. We also construct a non-constant periodic function $v_0$ such that its Fermi variety is reducible for  some $\lambda$, which disproves a conjecture by  Gieseker, Kn\"orrer and Trubowitz.

Under some assumptions of irreducibility of Fermi variety $F_{\lambda}(v_0)/\Z^d$, we show that $H=-\Delta +v_0+v$ does not have any embedded eigenvalues provided that $v$ decays  exponentially. The assumptions are conjectured to be true for any periodic function $v_0$. As an application, we show that when $d=2$, $H=-\Delta +v_0+v$ does not have any embedded eigenvalues provided that $v$ decays exponentially. 

 

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’'.

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