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

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.