Generalized Robin Boundary Conditions, Robin-to-Dirichlet Maps, and Krein-Type Resolvent Formulas for Schr\"odinger Operators on Bounded Lipschitz Domains

TL;DR

This paper develops advanced boundary condition frameworks and resolvent formulas for Schrödinger operators on Lipschitz and $C^{1,r}$-domains, enhancing mathematical tools for quantum and wave physics in irregular geometries.

Contribution

It introduces generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas specifically for Schrödinger operators on Lipschitz and $C^{1,r}$-domains, extending existing theories.

Findings

Established new Krein-type resolvent formulas for Lipschitz domains.

Extended boundary condition analysis to $C^{1,r}$-domains.

Provided mathematical tools for quantum mechanics in irregular geometries.

Abstract

We study generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schr\"odinger operators on bounded Lipschitz domains in $\bbR^n$, $n\ge 2$. We also discuss the case of bounded $C^{1,r}$-domains, $(1/2)<r<1$.