Resonant tunnelling through short-range singular potentials

Abstract

A three-parameter family of point interactions constructed from sequences of symmetric barrier–well–barrier and well–barrier–well rectangles is studied in the limit, when the rectangles are squeezed to zero width but the barrier height and the well depth become infinite (the zero-range limit). The limiting generalized potentials are referred to as the second derivative of Dirac's delta function ±λδ″(x) with a renormalized coupling constant λ > 0 or simply as ±δ″-like point interactions. As a result, a whole family of self-adjoint extensions of the one-dimensional Schrödinger operator is shown to exist, which results in full and partial resonant tunnelling through this class of singular potentials. The resonant tunnelling occurs for countable sets of interaction strength values in the λ-space which are the roots of several transcendental equations. The comparison with the previous results for δ′-like point interactions is also discussed.