Scattering in quantum networks

Simplest quantum network is a junction constructed as a union of a compact basic domain with a piecewise smooth boundary and few straight links of a constant width connecting the basic domain with infinity. We consider a one-body scattering problem for a Schödinger operator on a junction with real continuous potential vanishing on the links. The role of the unperturbed operator is played by the orthogonal sum of the corresponding Schrödinger operator on the basic domain and the Laplacian on the links. The basic difficulty of the problem is defined by presence of embedded eigenvalues of the unperturbed problem, which give rise to “dangerous resonances” when the zero boundary condition on the bottom sections of the links is replaced by the smooth matching condition. In a relevant problem of celestial mechanics, with small denominators, H. Poincare suggested elimination of the similar “dangerous resonances” as a preliminary step toward calculation of the solution of the corresponding Hamiltonian system by an analytic perturbation procedure. In the scattering problem on the thin quantum network a similar elimination is achieved via construction of a solvable model with the scattering matrix equal to the local Blaschke factor of the full Scattering matrix with a single resonance pole situated near to the Fermi level of the junction.