Supersymmetric quantum mechanics with a point singularity

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 δ( x)-potential. Similarly, each of the walls at 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.