Supersymmetric Quantum Mechanics under Point Singularities

TL;DR

This paper systematically explores supersymmetry in one-dimensional quantum systems with point singularities and boundaries, revealing conditions for unbroken SUSY and extending known models to more general singularities.

Contribution

It introduces a comprehensive framework for SUSY in quantum systems with general point singularities and boundaries, extending existing models and analyzing SUSY breaking conditions.

Findings

Certain parameter choices yield N=1 SUSY, which can be enhanced to N=4.

Interval systems can support N=2 or N=4 SUSY, with various spectral degeneracies.

Includes known SUSY systems with delta potentials as special cases.

Abstract

We provide a systematic study on the possibility of supersymmetry (SUSY) for one dimensional quantum mechanical systems consisting of a pair of lines $\R$ or intervals [-l, l] each having a point singularity. We consider the most general singularities and walls (boundaries) at $x = \pm l$ admitted quantum mechanically, using a U(2) family of parameters to specify one singularity and similarly a U(1) family of parameters to specify one wall. With these parameter freedoms, we find that for a certain subfamily the line systems acquire an N = 1 SUSY which can be enhanced to N = 4 if the parameters are further tuned, and that these SUSY are generically broken except for a special case. The interval systems, on the other hand, can accommodate N = 2 or N = 4 SUSY, broken or unbroken, and exhibit a rich variety of (degenerate) spectra. Our SUSY systems include the familiar SUSY systems with the Dirac $\delta(x)$-potential, and hence are extensions of the known SUSY quantum mechanics to those with general point singularities and walls. The self-adjointness of the supercharge in relation to the self-adjointness of the Hamiltonian is also discussed.