Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators

TL;DR

This paper derives trace formulae and estimates for the singular values of resolvent power differences of self-adjoint elliptic operators with non-local Robin boundary conditions, extending classical results and connecting boundary maps to spectral properties.

Contribution

It provides new Schatten--von Neumann estimates and trace formulae for resolvent power differences of elliptic operators with non-local Robin boundary conditions, generalizing previous classical estimates.

Findings

Schatten--von Neumann type estimates for singular values

Trace class property for resolvent power differences when m > n/2 - 1

Trace formulae linking resolvent differences to boundary maps

Abstract

In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain $\Omega\subset\dR^n$ with smooth compact boundary are studied. A Schatten--von Neumann type estimate for the singular values of the difference of the $m$th powers of the resolvents of two Robin realizations is obtained, and for $m>\tfrac{n}{2}-1$ it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M. Sh. Birman where one of the Robin realizations is replaced by the Dirichlet operator. In both cases trace formulae are proved, in which the trace of the resolvent power differences in $L^2(\Omega)$ is written in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps on the boundary space $L^2(\partial\Omega)$.