Resonance one-body scattering on a junction

In this paper we propose a synthesis of various approaches mixing computational modeling, solving complex and sometimes ill-posed inverse problems and the development of efficient analytic perturbation procedures, which offer an analytic path to the solution of the mathematical design and optimization problems for constructing quantum networks with prescribed transport properties. We consider the simplest sort of 2$D$ quantum networks – the junctions – and focus on the problems of the resonance scattering, caused by the spectral properties of the relevant Schrödinger operator on the vertex domain. Typically, 1-D features appear in the form of the single-mode scattering on the first spectral (energy) band in the resulting solvable model, but the analysis of multi-mode scattering is possible with our methodology. However, this comes at the price of assuming realistic (as opposed to quite general) matching between the scattering Ansatz in the wires and the solution of the Schrödinger equation on the vertex domain. Here this matching is based on a recently developed version of the Dirichlet-to-Neumann map. We are further able to observe the transformation of the discrete spectrum of the Schrödinger operator on the vertex domain into the resonance features of the relevant scattering problem.