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Supersymmetric Quantum Mechanics with a Point Singularity

Takashi Uchino·Izumi Tsutsui
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Abstract

We study the possibility of supersymmetry (SUSY) in quantum mechanics in one dimension under the presence of a point singularity. The system considered is the free particle on a line R or on the interval [-l, l] where the point singularity lies at x = 0. In one dimension, the singularity is known to admit a U(2) family of different connection conditions which include as a special case the familiar one that arises under the Dirac delta-potential. Similarly, each of the walls at x = l and x = -l admits a U(1) family of boundary conditions including the Dirichlet and the Neumann boundary conditions. Under these general connection/boundary conditions, the system is shown to possess an N = 1 or N = 2 SUSY for various choices of the singularity and the walls, and the SUSY is found to be `good' or `broken' depending on the choices made. We use the supercharge which allows for a constant shift in the energy, and argue that if the system is supersymmetric then the supercharge is self-adjoint on states that respect the connection/boundary conditions specified by the singularity.

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Takashi Uchino, Izumi Tsutsui. 2002-10-11. Supersymmetric Quantum Mechanics with a Point Singularity. https://doi.org/10.1016/s0550-3213(03)00352-3

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