Inverse scattering on perturbed lattices

Inverse Problems and Applications

22 April 14:00 - 15:00


We develop a spectral and inverse scattering theory on a class of perturbed periodic graph, which includes the square lattice, the triangluar lattice, the hexagonal lattice, the Kagome lattice and the diamond lattice. We consider the Laplacian on this graph, perturbed on a compact set by changing the graph structure or by adding a potential. We can then derive the resolvent estimates (limiting absorption principle), construct the generalized eigenfunctions describing the continuous spectrum. The Rellich type theorem holds, which shows that a solution of the Helmholtz equation outside a comact set decaying in a certain order vanishes near infinity. This allows us to prove the equivalence of the S-matrix and the Dirichlet-Neumann map for the boundary value problem in a bounded domain. For the case of the square lattice, it is proven that the potential is uniquely reconstructed from the S-matrix of a fixed energy with a finite algorithm of computation. This is carried out without any restriction of the space dimension (even for 2-dim.). The same results are expected for other cases.