Generalized boundary triples for adjoint pairs with applications to non-self-adjoint Schr\"odinger operators

TL;DR

This paper extends the concept of generalized boundary triples to adjoint pairs of operators, providing criteria for boundary parameters to induce closed operators with nonempty resolvent sets, with applications to complex Schrödinger operators.

Contribution

It introduces a generalized framework for boundary triples in the context of adjoint pairs and applies it to analyze non-self-adjoint Schrödinger operators with complex potentials.

Findings

Criteria for boundary parameters to produce closed operators with nonempty resolvent sets

Extension of boundary triple theory to adjoint pairs of operators

Application to Schrödinger operators with complex potentials on Lipschitz domains

Abstract

We extend the notion of generalized boundary triples and their Weyl functions from extension theory of symmetric operators to adjoint pairs of operators, and we provide criteria on the boundary parameters to induce closed operators with a nonempty resolvent set. The abstract results are applied to Schr\"odinger operators with complex $L^p$-potentials on bounded and unbounded Lipschitz domains with compact boundaries.