Contact interactions and Kronig-Penney models in Hermitian and PT symmetric quantum mechanics

The delta function potential is a simple model of zero-range contact interaction in non-relativistic quantum mechanics in one dimension. The Kronig-Penney model is a one-dimensional periodic array of delta functions and provides a simple illustration of 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 PT symmetric quantum mechanics. In conventional Hermitian quantum mechanics we determine the most general contact interaction compatible with self-adjointness and in PT quantum mechanics we consider interactions that respect symmetry under the transformation PT where P denotes parity and T denotes time reversal. In both cases we find that the most general interaction has four independent real parameters and depending on the values of those parameters the contact interaction can support zero, one or two bound states. By contrast the conventional delta function can only support zero or one bound state. In the PT symmetric case moreover the two bound state energies can be both real or a complex conjugate pair. The transition from real to complex bound state energies corresponds to the spontaneous breaking of PT symmetry. The scattering states for the PT symmetric case are also found to exhibit spontaneous breaking of PT symmetry wherein the eigenvalues of the non-unitary S-matrix depart the unit circle in the complex plane. We also 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 we find that 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.