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
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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 develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting "scattering matrix" mapping radiation fields on the past horizon and past null infinity to radiation fields on the future horizon and future null infinity is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity is bounded. On the frequency side, this corresponds to the novel statement that the suitably normalised reflection and transmission coefficients are uniformly bounded independently of the frequency parameters. We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity of suitable wave-packets as above. The results make essential use of a refinement of our recent proof [M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case |a|<M, arXiv:1402.6034] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards. The aim of the present work is to validate a full vectorial electromagnetic inverse scattering algorithm against experimental data. Data were provided courtesy of Institut Fresnel, Marseille, France. These data were carried out in an anechoic chamber and correspond to different canonical targets as well as one mysterious object which is known only by experimentalists who measured the associated scattered field. The inverse algorithm was first developed in the optical domain and is adapted herein to the microwave domain. It is an iterative approach where the parameter of interest, namely the relative permittivity distribution, is updated gradually by minimizing a cost function describing the discrepancy between data and those that would be obtained via a forward solver for the best available estimate of the relative permittivity. The forward solver is based on the coupled dipole method which was introduced in the seventies to study the scattering of light by non-spherical dielectric grains. The forward and inverse schemes are briefly described and various examples are presented that demonstrate the efficiency of the inverse algorithm. A new method, based on an iterative procedure, for solving the two-dimensional inverse scattering problem is presented. This method employs an equivalent Neumann series solution in each iteration step. The purpose of the algorithm is to provide a general method to solve the two-dimensional imaging problem when the Born and the Rytov approximations break down. Numerical simulations were calculated for several cases where the conditions for the first order Born approximation were not satisfied. The results show that in both high and low frequency cases, good reconstructed profiles and smoothed versions of the original profiles can be obtained for smoothly varying permittivity profiles (lossless) and discontinuous profiles (lossless), respectively. A limited number of measurements around the object at a single frequency with four to eight plane incident waves from different directions are used. The method proposed in this article could easily be applied to the three-dimensional inverse scattering problem, if computational resources are available.