Modified Krein Formula and Analytic Perturbation Procedure for Scattering on Arbitrary Junction

A quantum network is constructed of straight quantum wiresthe leads, of constant width,and quantum wells, which play roles of the vertex domains of the network. Basic element of the network is the junction: a detail of the network consisting of a single compact quantum well and few semi-infinite wires attached to it. In the theoretical study of the one-body transport on a junction, the role of the Hamiltonian is played by the one-body Schrödinger operator. In case when the corresponding potential is a real constant in the wires and is a piecewise continuous bounded real function on the quantum well, the transport problem is reduced to the one-body scattering problem. In this paper we suggest a semi-analytic perturbation procedure which permits to calculate the onebody scattering parameters for arbitrary junction, based on a specially selected intrinsic large parameter. This procedure gives us an approximate scattering matrix. The suggested analytic perturbation procedure is applicable to any junction based on a compact vertex domain, with piece-wise smooth boundary. Scattering matrix of a thin junction is approximated by the scattering matrix of the corresponding solvable model.