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Grigori Rozenblum: Spectral properties of the resolvent difference of Schrödinger type operators with singular perturbat

Date: 2025-05-27

Time: 09:15 - 10:15

Zoom link: https://kva-se.zoom.us/j/9217561890

Speaker
Grigori Rozenblum, Chalmers and University of Gothenburg

Abstract
There are three types of spectral problems having attracted much attention, concerning the difference of resolvents and powers of resolvents:
 
(1)  A  second order elliptic operator with Robin boundary conditions with two different weight functions;

(2) A second order elliptic operator with delta-potential on a Lipschitz surface with two different weight functions;

(3) A second order elliptic operator with two different potentials supported on a fractal subset.
 
It turns out that all these problems  fit into one abstract scheme and sharp eigenvalue estimates can be derived from recent results, joint with E.Shargorodsky and G.Tashchiyan, about Birman-Schwinger type operators with density being a singular measure.  Further, for  Problem (2), eigenvalue asymptotics can be derived from earlier results with G.Tashchyan about potential type operators on Lipshitz surfaces.  As for the asymptotics for  Problem (1), the application of  known results requires considerable smoothness of the boundary. To dispose of this smoothness, a further reduction is performed to a fourth order Steklov-type problem, where the recent perturbational approach  used by the author for the classical second order Steklov problem in Lipschitz domains can be adapted.