Scattering data and bound states of a squeezed double-layer structure
Abstract
A heterostructure composed of two parallel homogeneous layers is studied in the limit as their widths l1 and l2, and the distance between them r shrinks to zero simultaneously. The problem is investigated in one dimension and the squeezing potential in the Schr\"odinger equation is given by the strengths V1 and V2 depending on the layer thickness. A whole class of functions V1(l1) and V2(l2) is specified by certain limit characteristics as l1 and l2 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 Vj, lj, 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 δ'-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.