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 2𝐷 quantum networks — the junctions — and focus on the problems of the resonance scattering, caused by the spectral properties of the relevant Schrodinger 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 Schrodinger 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 Schrodinger operator on the vertex domain ¨ into the resonance features of the relevant scattering problem. 

We have developed a theory of the one-phonon intraband resonance scattering of electromagnetic radiation (IRSER) in anisotropic quantum dots subjected to an arbitrarily directed magnetic field. The differential cross section of scattering is obtained. The resonance structure of the cross section is studied. It is shown that the quantum dot subjected in a magnetic field can be used as the detector of phonon modes. The interesting multiplet structure of the resonance peaks is studied.