On Compressed Resolvents of Schrödinger Operators with Complex Potentials

Abstract

The compression of the resolvent of a non-self-adjoint Schrödinger 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.

arXiv commentto appear in the Henk de Snoo 75th birthday topical issue in Complex Analysis and Operator Theory