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).
There is an extensive literature on the mathematical theory of graphs and their spectra. We cite here only the articles which have close relations to this paper, but are not mentioned above. For a general survey, see e.g. @cite_34 and the references therein.
@cite_34
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Introduction. Operateurs lineaires associes a un graphe. Resultats fondamentaux. Rayon spectral, fonctions generatrices de marche et mesures spectrales. Croissance et nombre isoperimetrique d'un graphe. Fonctions propres positives. Graphes de groupes, arbres et graphes reguliers en distance. Quelques remarques sur les applications en chimie et en physique