Quantum Switch is one of most important elements of quantum networks, see, for instance [53, 54, 55]. In the present paper mathematical modeling of the Resonance Quantum Switch is reduced to the solution of the corresponding Scattering Problem on a network consisting of a quantum well and few quantum wires attached to it. In a simplest practical problem the roles of quantum wires are played by straight rectangular channels and the role of the quantum well is played by a circular domain, with the boundary having no flat pieces to match the rectangular channels. We suggest a procedure of matching the rectangular channels and a general convex domain with piecewise smooth boundary. The matching problem is reduced to some finite-dimensional equation with matrix-functions. Important spectral characteristics of the above scattering problem such as positions and life-times of resonances may be found numerically from the corresponding determinant condition or via asymptotic analysis — for sufficiently narrow channels.