Scattering on a Compact Domain with Few Semi-Infinite Wires Attached: Resonance Case

A Scattering problem is studied for Neumann Laplacian with a continuous potential on a domain with a smooth boundary and few semi-infinite wires attached to it at the points of contact on the boundary. In resonance case when the frequency of the incoming waves in the wires coincides with some resonance frequency of the domain the approximate formula for the transmission coefficient from one wire to another is derived: in the case of weak interaction between the domain and the wires the transmission coefficient is proportional to the product of values of the corresponding resonance eigenfunction of inner problem at the corresponding points of contact.