We study the inverse scattering for Schrodinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph, and reconstruct scalar potentials as well as the graph structure from the knowledge of the S-matrix. In particular, we give a procedure for probing defects in hexagonal lattices (graphene).
The inverse scattering for the multi-dimensional discrete Schr "o dinger operator was first reported in @cite_21 . In @cite_17 , it was proven by using the complex Born approximation of the scattering amplitude. The extension to the hexagonal lattice is done in @cite_6 . The Rellich type theorem for the uniqueness of solutions to the Helmholtz equation, proved in @cite_38 , plays an essential role in this paper. The Hilbert Nullstellensatz is used in the proof and this idea goes back to Shaban-Vainberg @cite_7 . The long-range scattering is discussed in @cite_20 .
@cite_38 @cite_7 @cite_21 @cite_6 @cite_20 @cite_17
2163901730 2009204289 2072789180 2025514457 2041224997
An analogue of Rellich's theorem is proved for discrete Laplacians on square lattices, and applied to show unique continuation properties on certain domains as well as non-existence of embedded eigenvalues for discrete Schrodinger operators. The problem of determining a unique solution of the Schrodinger equation on the lattice is considered, where Δ is the difference Laplacian and both f and q have finite supports. It is shown that there is an exceptional set So of points on for which the limiting absorption priciple fails, even for unperturbed operator (q(x)=0). This exceptional set consists of the points when d is even and when d is odd. For all Values of , the radiation conditions are found which single out the same solutions of the problem as the ones determined by the limiting absorption principle. These solutions are conbinations of several waves propagating with different frequencies, and the number of waves depends on the value of λ. We consider the spectral theory and inverse scattering problem for discrete Schrodinger operators on the hexagonal lattice. We give a procedure for reconstructing finitely supported potentials from the scattering matrices for all energies. The same procedure is applicable for the inverse scattering problem on the triangle lattice. We consider the scattering theory for discrete Schrodinger operators on Zd with long-range potentials. We prove the existence of modified wave operators constructed in terms of solutions of a Hamilton-Jacobi equation on the torus Td. We study discrete Schrodinger operators with compactly supported potentials on Z d . Constructing spectral representations and representing S-matrices by the generalized eigenfunctions, we show that the potential is uniquely reconstructed from the S-matrix of all energies. We also study the spectral shift function ( ( ) ) for the trace class potentials, and estimate the discrete spectrum in terms of the moments of ( ( ) ) and the potential.