Self-adjoint extensions and SUSY breaking in Supersymmetric Quantum Mechanics

TL;DR

This paper investigates how different self-adjoint extensions of supercharges in SUSY quantum mechanics affect the realization of N=2 supersymmetry, revealing conditions for unbroken and broken SUSY states.

Contribution

It characterizes the self-adjoint extensions that preserve N=2 SUSY and distinguishes them from those leading to only N=1 SUSY with broken symmetry.

Findings

Two specific scale-invariant extensions realize N=2 SUSY.

Most extensions result in only N=1 SUSY with broken symmetry.

Energy spectra are non-degenerate in broken SUSY cases.

Abstract

We consider the self-adjoint extensions (SAE) of the symmetric supercharges and Hamiltonian for a model of SUSY Quantum Mechanics in $\mathbb{R}^+$ with a singular superpotential. We show that only for two particular SAE, whose domains are scale invariant, the algebra of N=2 SUSY is realized, one with manifest SUSY and the other with spontaneously broken SUSY. Otherwise, only the N=1 SUSY algebra is obtained, with spontaneously broken SUSY and non degenerate energy spectrum.