Supersymmetric Descendants of Self-Adjointly Extended Quantum Mechanical Hamiltonians

TL;DR

This paper explores the self-adjoint extensions of Hamiltonians in supersymmetric quantum mechanics, characterizing their descendants, and analyzing concrete examples like particles in a box with various boundary conditions.

Contribution

It provides a detailed classification of self-adjoint extensions and their supersymmetric descendants, including explicit examples and analysis of boundary conditions and resonances.

Findings

Only a 3-parameter sub-family of extensions has supersymmetric self-adjoint descendants.

Explicit examples include particles in a box with general boundary conditions.

Bulk-boundary resonances are discussed in the context of supersymmetry.

Abstract

We consider the descendants of self-adjointly extended Hamiltonians in supersymmetric quantum mechanics on a half-line, on an interval, and on a punctured line or interval. While there is a 4-parameter family of self-adjointly extended Hamiltonians on a punctured line, only a 3-parameter sub-family has supersymmetric descendants that are themselves self-adjoint. We also address the self-adjointness of an operator related to the supercharge, and point out that only a sub-class of its most general self-adjoint extensions is physical. Besides a general characterization of self-adjoint extensions and their supersymmetric descendants, we explicitly consider concrete examples, including a particle in a box with general boundary conditions, with and without an additional point interaction. We also discuss bulk-boundary resonances and their manifestation in the supersymmetric descendant.