Point interactions with bound states: A zero-thickness limit of a double-layer heterostructure

Abstract

A heterostructure composed of two parallel homogeneous layers is studied in the limit as their width and the distance between them shrinks to zero simultaneously. The problem is considered in one dimension and the squeezing potential in the Schrödinger equation is chosen in the form of a piecewise constant function. As a result, two families of point interactions with bound state energy are realized from this structure. The specific feature of these interactions is the resonant-tunneling transmission of electrons through one-point singular potentials under certain conditions described by transcendental equations. The solutions to these equations define so-called resonance sets of Lebesgue’s measure zero. A particular example is the potential in the form of the derivative of Dirac’s delta function. For a whole family of point interactions including this example, the existence of a bound state is proven, contrary to the widespread opinion on the non-existence of bound states in δ'-like systems.