Analysis of a Stratified Quantum Waveguide with Interactions at Interface Planes

In this paper we consider a quantum waveguide that consists of three strata ∏0 = {(x, x 3) ∈ 3 : x 3 < 0}, ∏0,h = {(x, x 3) ∈ 3 : 0 < x 3 < h }, ∏ h = {(x, x 3) ∈ 3 : x 3 > h }, where x = (x 1, x 2) ∈ 2. A potential of the form q = qr + qs is established in this structure, where qr is a regular bounded potential depending on only the coordinate x 3, and qs is the singular potential qs = α 1 δ (x 3) + β 1 δ (x 3) + α 2 δ (x 3 - h) + β 2 δ (x 3 - h) with support at the planes x 3 = 0 and x 3 = h. The Green's function of the waveguide is constructed as an expansion involving the eigenfunctions and generalized eigenfunctions of an auxiliary one-dimensional Schrödinger operator. The asymptotic analysis of the Green's function is carried out by means of the stationary phase method. This gives the leading contribution of the Green's function far from the point source. Finally some numerical examples are considered for the application of the present analysis.