Schatten-von Neumann estimates for resolvent differences of Robin Laplacians on a half-space

TL;DR

This paper investigates the Schatten-von Neumann class properties of resolvent differences of Robin Laplacians on a half-space, showing compactness and class membership under certain boundary coefficient conditions.

Contribution

It establishes conditions under which the resolvent difference of Robin Laplacians on a half-space belongs to Schatten-von Neumann classes, extending known results to unbounded domains.

Findings

Resolvent difference is compact under certain boundary coefficient conditions.

The resolvent difference belongs to Schatten-von Neumann classes if coefficients are sufficiently close.

Results extend Schatten-von Neumann class membership to half-space Robin Laplacians.

Abstract

The difference of the resolvents of two Laplacians on a half-space subject to Robin boundary conditions is studied. In general this difference is not compact, but it will be shown that it is compact and even belongs to some Schatten-von-Neumann class, if the coefficients in the boundary condition are sufficiently close to each other in a proper sense. In certain cases the resolvent difference is shown to belong even to the same Schatten-von Neumann class as it is known for the resolvent difference of two Robin Laplacians on a domain with a compact boundary.