Complete Non-Selfadjointness for Schr\"odinger Operators on the Semi-Axis

TL;DR

This paper characterizes the complete non-selfadjointness of maximally dissipative extensions of Schrödinger operators on a half-line, revealing conditions under which these extensions are or are not selfadjoint.

Contribution

It provides a complete characterization of maximally dissipative extensions of Schrödinger operators regarding their non-selfadjointness and identifies conditions for selfadjoint reducing subspaces.

Findings

All extensions preserving the differential expression are completely non-selfadjoint.

Extensions can have one-dimensional selfadjoint reducing subspaces.

A specific example illustrating these properties is provided.

Abstract

In this note we investigate complete non-selfadjointness for all maximally dissipative extensions of a Schr\"odinger operator on a half-line with dissipative bounded potential and dissipative boundary condition. We show that all maximally dissipative extensions that preserve the differential expression are completely non-selfadjoint. However, it is possible for maximally dissipative extensions to have a one-dimensional reducing subspace on which the operator is selfadjoint. We give a characterisation of these extensions and the corresponding subspaces and present a specific example.