Supersymmetric quantum mechanics with a point singularity

Abstract

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 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. *

Supersymmetry in quantum mechanics with point interactions

“…(3.17) and (3.20)). 10 Thus, our results give an extension of the work [16] for a free particle to a supersymmetric system with a superpotential. It should be stressed that although the supercharges (5.7) could be represented by 2 × 2 matrices, as done above for a special case, our representation of the supercharges has the advantage of clarifying the role of the discrete transformations P j .…”

“…Then, a point singularity associated with g = σ 3 describes a perfect wall through which no probability flow can penetrate, so that the circle S 1 can be regarded as an interval [−l, l] with a point singularity at x = 0 in ref. [16]. We further need to restrict the superpotential to be constant (W ′ (x) = −W ′ (−x) = b for 0 < x < l), in order to have the free Hamiltonian.…”

“…(5.7) on Ψ(x) can be represented by the 2 × 2 matrices: The above representation of the supercharges agrees with that of ref. [16] (see eqs. (2.21)), up to normalization.…”

“…The allowed connection conditions (5.9)-(5.12) are also found in ref. [16] (see eqs. (3.17) and (3.20)).…”

“…The degeneracy results from N = 4 supersymmetry and we can construct four hermite supercharges. An easy way to represent the supercharges is to introduce, as a natural extension of the work [16], a four-component wavefunction Ψ(…”