TITLE: Localization and spectral statistics for Schr\"odinger operators with random point interactions

Talks take place at 11:00-12:00 noon, Tuesday in POT 745, unless otherwise noted. For updates and more information, please visit: http://ms.uky.edu/~mhto224/seminar/PDEseminarS2018.html

If you have any questions about a particular seminar, please contact the speaker's host. For general questions about the seminar, please contact Mihai Tohaneanu using the email address mihai.tohaneanu@uky.edu. 

The Bernstein Problem in Two Dimensions

I will outline a complete solution to the weighted approximation problem for polynomials

on an arbitrary bounded simply connected domain  in the complex plane. In that setting

the problem was first studied extensively by Keldysh prior to 1941 in the context of L2-

approximation, and more than four decades later by Beurling where, in the latter instance,

the emphasis was on uniform approximation. Here, Beurling obtains the sharper result

with respect to the weight w, but at the expense of limiting the type of region to which

his argument applies. Ironically, however, the two problems turned out to be essentially

equivalent, but neither Beurling nor Keldysh obtained what might be considered a definitive

solution. My presentation will focus on the L2-case, where the theory of Sobolev spaces and

its associated potential theory is available. It is a simple matter to pass from there to a

solution in the case of uniform approximation.

Title: Compactness of iso-resonant potentials for Schrodinger operators on R^d

Abstract: In joint work with R.\ Wolf, we prove compactness of a restricted set of real-valued, compactly supported potentials $V$ for which the corresponding Schr\"odinger operators $H_V$ have the same resonances, including multiplicities. More specifically, let $B_R(0)$ be the ball of radius $R > 0$ about the origin in $\R^d$, for $d=1,3$. Let $\mathcal{I}_R (V_0)$ be the set of real-valued potentials in $C_0^\infty( \overline{B}_R(0); \R)$ so that the corresponding Schr\"odinger operators have the same resonances, including multiplicities, as $H_{V_0}$. We prove that the set $\mathcal{I}_R (V_0)$ is a compact subset of $C_0^\infty (B_R(0))$ in the $C^\infty$-topology.