The spectrum of two quantum layers coupled by a window

Abstract

We consider the Dirichlet Laplacian in a domain two three-dimensional parallel layers having common boundary and coupled by a window. The window produces the bound states below the essential spectrum; we obtain two-sided estimates for them. It is also shown that the eigenvalues emerge from the threshold of essential spectrum as the window passes through certain critical shapes. We prove the necessary condition for the window to be of critical shape. Under an additional assumption we show that this condition is sufficient and obtain the asymptotic expansion for the emerging eigenvalue as well as for the associated eigenfunction.

Perturbation of Threshold of Essential Spectrum for Waveguides with Windows. II: Asymptotics

“…As is shown in [1,2], the fact that a window is of critical shape is equivalent to the existence of a nontrivial bounded solution to the problem (2.1). Assumption (A) covers a typical situation.…”

“…Outside some neighborhood of the curve ∂ω ×{0}, the replacement x → y is identical. This change of variables was described in [1, Section 7, formula (7.1)] and [2,Section 7].…”

“…Each of these objects is associated with its proper spectral characteristic located near the threshold of the essential spectrum. The leading terms of asymptotic expansions of such characteristics were found in [1] and also in [2] in the case of bound states. However, the technique of [1,2] does not work for obtaining complete asymptotic expansions, and this question is still open.…”

“…It is shown [1,2] that there exist windows of critical shape, i.e., such that there is a virtual level on the threshold of the essential spectrum, which means that bound and anti-bound states or pairs of resonance states emerge from the threshold of the essential spectrum. The exact definitions are given in Section 2.…”

“…Similar results were established in the case of dimension equal to or greater than 3 (cf. [1,2,12,13], [17]- [20]). We emphasize the paper [4] presenting an original approach for proving the existence and an analysis of the asymptotic behavior of the egenvalues.…”