Joint Analysis/Applied Math seminar-Green's functions and the existence of the gauge for local and non-local time-independent Schrodinger equations

Joint Analysis/Applied Math seminar
Date: Wednesday, November 29, 2017
Time: 11am-12noon
Place: Rome 204
 
Speaker: Michael Frazier, University of Tennessee

Title:  Green's functions and the existence of the gauge for local and
non-local time-independent Schrodinger equations

Abstract:  We discuss a series of papers with Igor Verbitsky (and in one case, with Fedor Nazarov), dealing with time-independent Schrodinger operators L = - Laplacian -q, where the potential q is non-negative.  The "gauge," as it is known in the probability literature, is the solution of Lu=0 on a domain, with u=1 on the boundary of the domain.  We obtain estimates for the Green's function of L, and conditions for the existence of the gauge, under very general conditions on the potential q.  The conditions, which describe how rapidly q can blow up near the boundary, are close to being necessary.  We also discuss related results when the Laplacian is replaced
by the fractional Laplacian.  The conclusions are based on a general estimate for the kernel of a Neumann series (I-T)^{-1} associated with an integral operator T of norm less than 1.