2D waveguides: asymptotics of eigenvalue induced by a window in a semitransparent separating wall

Abstract

Two-dimensional quantum waveguides with a common semitransparent wall are considered. It is assumed that there is a small window in the separating wall. It leads to the appearance of an eigenvalue below the continuous spectrum. A semitransparent wall is introduced as a potential supported by a hypersurface. We use the method of asymptotic expansions of boundary problems solutions. It allows us to obtain the main term of the asymptotics explicitly.