Session S38 - Geometric Potential Analysis

Friday, July 16, 17:50 ~ 18:20 UTC-3

Positive solutions to the time-independent Schrodinger equation and the existence of the gauge

Michael Frazier

University of Tennessee, United States of America   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We survey some joint work with Igor Verbitsky and Fedor Nazarov regarding the time-independent Schrodinger equation $-\triangle u= \omega u$ on a domain $\Omega$, $u=f$ on $\partial \Omega$. Our goal is to find minimal conditions on the Schrodinger potential $\omega$ that guarantee the existence of a solution $u \geq 0$ when $f \geq 0$. We begin with a general result about quasi-metric kernels, which yields matching upper and lower bounds for Green's function of the Schrodinger operator. A condition for the existence of the gauge (the solution $u$ if $f=1$) for Schrodinger operators is given in terms of the exponential integrability of the balayage of $\omega$. Recently, these results were extended to the case of uniform domains, with Martin's kernel replacing the Poisson kernel. We also discuss related results for the fractional Laplacian.

Joint work with Fedor Nazarov (Kent State University, United States of America) and Igor Verbitsky (University of Missouri, United States of America).

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