On Compressed Resolvents of Schrödinger Operators with Complex Potentials

Abstract

The compression of the resolvent of a non-self-adjoint Schrödinger operator $$-\Delta +V$$ - Δ + V onto a subdomain $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n is expressed in a Kreĭn–Naĭmark type formula, where the Dirichlet realization on $$\Omega $$ Ω , the Dirichlet-to-Neumann maps, and certain solution operators of closely related boundary value problems on $$\Omega $$ Ω and $${\mathbb {R}}^n\setminus {\overline{\Omega }}$$ R n \ Ω ¯ 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.