Scattering on graphs and one-dimensional approximations to N-dimensional Schrödinger operators

Abstract

In the present article we develop the spectral analysis of Schrödinger operators on lattice-type graphs. For the basic example of a cubic periodic graph the problem is reduced to the spectral analysis of certain regular differential operators on a fundamental star-like subgraph with a selfadjoint condition at the central node and quasiperiodic conditions at the boundary vertices. Using an explicit expression for the resolvent of lattice-type operator we develop in the second section appropriate Lippmann-Schwinger techniques for the perturbed periodic operator and construct the corresponding scattering matrix. It serves as a base for the approximation of the multi-dimensional Schrödinger operator by a one-dimensional operator on the graph: in the third section of the paper for given N-dimensional Schrödinger operators with rapidly decreasing potential we construct a lattice-type operator on a cubic graph embedded into RN and show that the original N-dimensional scattering problem can be approximated in a proper sense by the corresponding scattering problem for the perturbed lattice operator.