Supersymmetry in quantum mechanics with point interactions

Abstract

We investigate supersymmetry in one-dimensional quantum mechanics with point interactions. We clarify a class of point interactions compatible with supersymmetry and present N = 2 supersymmetric models on a circle with two point interactions as well as a superpotential. A hidden su(2) structure inherent in the system plays a crucial role to construct the N = 2 supercharges. Spontaneous supersymmetry breaking due to point interactions and an extension to higher N-extended supersymmetry are also discussed.

Supersymmetry and discrete transformations on S1 with point singularities

“…Adding P n+1 to the algebra, we can construct 2n + 1 supercharges that form an N = 2n + 1 superalgebra [13]. Any subset of the 2n + 1 supercharges does not, however, coincide with the N = 2n supercharges in Eqs.…”

“…Thus, the N = 2n + 1 supersymmetry including P n+1 in the algebra belongs to a different class from the N = 2n supersymmetry considered in this Letter. Full details will be discussed in a forthcoming paper [13]. …”

“…A point singularity is parameterized by the group U(2) [1,2,3], and the parameters characterize connection conditions between a wavefunction and its derivative at the singularity. The variety of the connection conditions leads to various interesting physical phenomena, such as duality [4,5], the Berry phase [6,7], scale anomaly [8] and supersymmetry [9,10,11,12]. Since a system with a number of point singularities possesses a wider parameter space, the system can have new features.…”

“…[11] to higher N-extended supersymmetry, we follow the approach of Ref. [10] to realize N = 2n supersymmetry for any integer n. A key ingredient in the analysis of Ref. [10] is a set of the discrete transformations that forms an su(2) algebra of spin 1/2.…”

Self-adjoint extensions and SUSY breaking in supersymmetric quantum mechanics

“…We should mention that self-adjoint extensions of supercharges and Hamiltonian for the SUSYQM of the free particle with a point singularity in the line and the circle have been considered in [21,22,23,24], where N = 1, 2 realization of SUSY are described. They have also been considered in the framework of the Landau Hamiltonian for two-dimensional particles in nontrivial topologies in [25] (see also [26]).…”