Waveguides coupled through a semitransparent barrier: a Birman-Schwinger analysis

TL;DR

This paper analyzes the spectral properties of coupled waveguides separated by a semitransparent barrier using Birman-Schwinger theory, revealing conditions for bound states and deriving asymptotic expansions for eigenvalues.

Contribution

It develops a Birman-Schwinger framework for a generalized Schrödinger operator modeling coupled waveguides with a semitransparent boundary, including weak-coupling expansions and bounds on bound states.

Findings

Bound states occur when the barrier is locally more transparent.

Derived weak-coupling expansion for the ground-state eigenvalue.

Provided an upper bound on the number of bound states.

Abstract

The paper is devoted to a model of a mesoscopic system consisting of a pair of parallel planar waveguides separated by an infinitely thin semitransparent boundary modeled by a transverse delta interaction. We develop the Birman-Schwinger theory for the corresponding generalized Schroedinger operator. The spectral properties become nontrivial if the barrier coupling is not invariant with respect to longitudinal translations, in particular, there are bound states if the barrier is locally more transparent in the mean and the coupling parameter reaches the same asymptotic value in both directions along the guide axis. We derive the weak-coupling expansion of the ground-state eigenvalue for the cases when the perturbation is small in the supremum and the L^1-norms. The last named result applies to the situation when the support of the leaky part shrinks: the obtained asymptotics differs from that of a double guide divided by a pierced Dirichlet barrier. We also derive an upper bound on the number of bound states.