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.
We suggest a method of fitting of a zero-range model of a tectonic plate under a boundary stress on the basis of comparison of the theoretical formulae for the corresponding eigenfunctions/eigenvalues with the results extraction under monitoring, in
the remote zone, of non-random (regular) oscillations of the Earth with periods 0.2-6 hours, on the background seismic process, in case of low seismic activity. Observations of changes of the characteristics of the oscillations (frequency, amplitude and polarization) in course of time, together with the theoretical analysis of the fitted model, would enable us to localize the stressed zone on the boundary of the plate and estimate the risk of a powerful earthquake at the zone.
