Scattering data and bound states of a squeezed double-layer structure

TL;DR

This paper analyzes the scattering data and bound states of a double-layer quantum heterostructure as it is squeezed to a point, revealing conditions for bound state existence even in delta-prime-like potentials.

Contribution

It introduces a novel analysis of the squeezing limit for double-layer structures, establishing conditions for the existence of bound states in the limit, including delta-prime potentials.

Findings

Existence of scattering data limits under specific parameter conditions.

Identification of two resonance sets where divergences cancel.

Proof of bound states surviving in the squeezed limit, including delta-prime potentials.

Abstract

A heterostructure composed of two parallel homogeneous layers is studied in the limit as their widths $l_1$ and $l_2$, and the distance between them $r$ shrinks to zero simultaneously. The problem is investigated in one dimension and the squeezing potential in the Schr\"{o}dinger equation is given by the strengths $V_1$ and $V_2$ depending on the layer thickness. A whole class of functions $V_1(l_1)$ and $V_2(l_2)$ is specified by certain limit characteristics as $l_1$ and $l_2$ tend to zero. The squeezing limit of the scattering data $a(k)$ and $b(k)$ derived for the finite system is shown to exist only if some conditions on the system parameters $V_j$, $l_j$, $j=1,2$, and $r$ take place. These conditions appear as a result of an appropriate cancellation of divergences. Two ways of this cancellation are carried out and the corresponding two resonance sets in the system parameter space are derived. On one of these sets, the existence of non-trivial bound states is proven in the squeezing limit, including the particular example of the squeezed potential in the form of the derivative of Dirac's delta function, contrary to the widespread opinion on the non-existence of bound states in $\delta'$-like systems. The scenario how a single bound state survives in the squeezed system from a finite number of bound states in the finite system is described in detail.