Scattering on graphs and one-dimensional approximation of $N-$dimensional Schr"odinger operators

Y. Melnikov, B.Pavlov


In actual paper we develop the spectral analysis of Schr"odinger
operators on lattice
type graphs. For basic example of qubic periodic graph the problem is
reduced to
the spectral analysis of the regular differential operators on a fundamental
subgraph with a selfadjoint condition at the central node and
conditions at the boundary vertices.
Using an explicite expression for resolvent of lattice-type
operator we develop in the second sections the
Lippmann- Schwinger techniques for
the perturbed periodic operator and construct the
corresponding scattering matrix. It serves as a
base for the approximation of the multy-dimensional Schr"odinger
operator by the
operator on graph : in the third section of the paper for given
$N$-dimensional Schr"odinger operators with rapidly decreasing
potential we construct a lattice-type operator on cubic graph
embedded into
${bf R}^N$ and show that the original $N$-dimensional scattering
can be approximated
in proper sense by the corresponding scattering problem for the
perturbed lattice

Lattice graphs, Krein formula, Scattering

Math Review Classification

Last Updated

36 p

This article is available in: