A phenomenon of splitting resonant-tunneling one-point interactions

TL;DR

This paper explores how a family of resonant-tunneling point interactions can be derived from thin-layer heterostructures, revealing a complex structure of resonance sets that explains previous discrepancies in transparency properties.

Contribution

It introduces a generalized family of $oldsymbol{ ext{delta}}'$-interactions with resonance sets forming continuous curves, connecting Kurasov's theory to physical tunneling phenomena.

Findings

Resonance sets form countable continuous curves.

Splitting of resonance sets influences tunneling transparency.

Connection established between mathematical models and physical heterostructures.

Abstract

The so-called $\delta'$-interaction as a particular example in Kurasov's distribution theory developed on the space of discontinuous (at the point of singularity) test functions, is identified with the diagonal transmission matrix, {\em continuously} depending on the strength of this interaction. On the other hand, in several recent publications, the $\delta'$-potential has been shown to be transparent at some {\em discrete} values of the strength constant and opaque beyond these values. This discrepancy is resolved here on the simple physical example, namely the heterostructure consisting of two extremely thin layers separated by infinitesimal distance. In the three-scale squeezing limit as the thickness of the layers and the distance between them simultaneously tend to zero, a whole variety of single-point interactions is realized. The key point is the generalization of the $\delta'$-interaction to the family for which the resonance sets appear in the form of a countable number of continuous two-dimensional curves. In this way, the connection between Kurasov's $\delta'$-interaction and the resonant-tunneling point interactions is derived and the splitting of the resonance sets for tunneling pla ys a crucial role.