A solvable model for scattering on a junction and a modified analytic perturbation procedure

We consider a one-body spin-less electron spectral problem for a resonance scattering system constructed of a quantum well weakly connected to a noncompact exterior reservoir, where the electron is free. The simplest kind of the resonance scattering system is a quantum network, with the reservoir composed of few disjoint cylindrical quantum wires, and the Schr\"{o}dinger equation on the network, with the real bounded potential on the wells and constant potential on the wires. We propose a Dirichlet-to-Neumann - based analysis to reveal the resonance nature of conductance across the star-shaped element of the network (a junction), derive an approximate formula for the scattering matrix of the junction, construct a fitted zero-range solvable model of the junction and interpret a phenomenological parameter arising in Datta-Das Sarma boundary condition, see {\cite{DattaAPL}, for T-junctions. We also propose using of the fitted zero-range solvable model as the first step in a modified analytic perturbation procedure of calculation of the corresponding scattering matrix.