Contact interactions and Kronig-Penney Models in Hermitian and PT-symmetric Quantum Mechanics

Abstract

The delta function potential is a simple model of zero-range contact interaction in one dimension. The Kronig-Penney model is a one-dimensional periodic array of delta functions that models the energy bands in a crystal. Here we investigate contact interactions that generalize the delta function potential and corresponding generalizations of the Kronig-Penney model within conventional and parity-time symmetric quantum mechanics (PTQM). In conventional quantum mechanics we determine the most general contact interaction compatible with self-adjointness; in PTQM we consider interactions that are symmetric under the combined transformation PT. In both cases we find that the most general interaction has four independent real parameters; depending on the parameter values the interaction can support more bound states than the conventional delta function. In the PT case the two bound state energies can be both real or a complex conjugate pair, with the transition corresponding to the breaking of PT-symmetry. The scattering states for the PT case are also found to exhibit spontaneous breaking of PT-symmetry. We investigate the energy bands when the generalized contact interactions are repeated periodically in space in one dimension. In the Hermitian case we find that the two bound states result in two narrow bands generically separated by a gap. These bands intersect at a single point in the Brillouin zone as the interaction parameters are varied. Near the intersection the bands form a massless Dirac cone. In the PT-symmetric case, as the parameters of the contact interaction are varied the two bound state bands undergo a PT-symmetry breaking transition wherein the two band energies go from being real to being a complex conjugate pair. The PT-symmetric Kronig-Penney model provides a simple soluble example of the transition which has the same form as in other models of PT-symmetric crystals.