Scattering in the case of the discrete Schrodinger operator for intersected quantum wires
The paper considers the discrete Schrodinger operator on a graph with vertices on two intersecting lines, which is perturbed by a decreasing potential. This operator is the Hamiltonian of an electron near a structure formed by a quantum dot and four outgoing quantum wires in the tight-binding approximation widely used in the physics literature for studying such nanostructures. We have proved the existence and uniqueness of the solution of the corresponding Lippmann-Schwinger equation and obtained the asymptotic formula for it. The non-stationary scattering picture has been studied. The scattering problem for the above operator in the case of a small potential, and also in the case of both a small potential and small velocity of a quantum particle, is investigated. Asymptotic formulas for the probabilities of the particle propagation in all possible directions have been obtained.