Singular Values and Trace Formulae for Resolvent Power Differences of Self-Adjoint Elliptic Operators
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Beschreibung
In the present thesis a unified functional analytic approach to the treatment of self-adjoint elliptic operators with Dirichlet, Neumann, Robin, and more general self-adjoint boundary conditions on bounded and unbounded domains is provided. Moreover, Schrödinger operators on couplings of exterior and interior domains with transmission boundary conditions are considered. In particular, Schrödinger operators with delta'-interactions on hypersurfaces are rigorously introduced. The key results in the thesis are Schatten-von Neumann estimates for the resolvent power differences of self-adjoint elliptic operators corresponding to the same differential expression and to distinct boundary conditions. Schatten-von Neumann estimates for the resolvent power differences of elliptic operators have a long history, starting in the middle of the 20th century with the seminal contributions by Povzner and Birman, followed by Grubb. In this thesis certain new estimates with faster convergence of singular values are obtained. The proofs of these estimates rely on Krein-type resolvent formulas, asymptotics of eigenvalues of the Laplace-Beltrami operator on the boundary and certain considerations of algebraic nature. A question of special interest, in connection with scattering theory, is the trace class property of the analyzed resolvent power differences, which implies the existence and completeness of the wave operators. In the special case, that the resolvent power differences are in the trace class, formulae for their traces are given.
Autor*in
Vladimir Lotoreichik
Themen in »Singular Values and Trace Formulae for Resolvent Power Differences of Self-Adjoint Elliptic Operators«
Elliptic operators Schatten-von Neumann classes singular values spectral estimates wave operators