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

In the present article we develop the spectral analysis of Schrodinger oper ators on lattice-type graphs. For the basic example of a cubic periodic gra ph the problem is reduced to the spectral analysis of certain regular diffe rential operators on a fundamental star-like subgraph with a selfadjoint co ndition at the central node and quasiperiodic conditions at the boundary ve rtices. Using an explicit expression for the resolvent of lattice-type oper ator we develop in the second section appropriate Lippmann-Schwinger techni ques for the perturbed periodic operator and construct the corresponding sc attering matrix. It serves as a base for the approximation of the multi-dim ensional Schrodinger operator by a one-dimensional operator on the graph: i n the third section of the paper for given N-dimensional Schrodinger operat ors with rapidly decreasing potential we construct a lattice-type operator on a cubic graph embedded into R-N and show that the original N-dimensional scattering problem can be approximated in a proper sense by the correspond ing scattering problem for the perturbed lattice operator. (C) 2001 America n Institute of Physics.