On Compressed Resolvents of Schr\"odinger Operators with Complex Potentials

TL;DR

This paper develops a Krein-Naimark type formula for the compressed resolvent of non-self-adjoint Schrödinger operators with complex potentials, linking boundary operators and solution operators in a boundary value problem framework.

Contribution

It introduces a novel formula for compressed resolvents of non-self-adjoint Schrödinger operators using boundary operators and extends coupling methods to complex potentials.

Findings

Derived a Krein-Naimark type formula for non-self-adjoint Schrödinger operators

Connected boundary value problems with operator theory in a new framework

Extended coupling methods to complex potential scenarios

Abstract

The compression of the resolvent of a non-self-adjoint Schr\"odinger operator $-\Delta+V$ onto a subdomain $\Omega\subset\mathbb R^n$ is expressed in a Krein-Naimark type formula, where the Dirichlet realization on $\Omega$, the Dirichlet-to-Neumann maps, and certain solution operators of closely related boundary value problems on $\Omega$ and $\mathbb R^n\setminus\overline\Omega$ are being used. In a more abstract operator theory framework this topic is closely connected and very much inspired by the so-called coupling method that has been developed for the self-adjoint case by Henk de Snoo and his coauthors.