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

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