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).
For the foundations of the properties of graph Laplacian, see @cite_16 and @cite_19 . A general approach to the spectral properties of periodic systems in terms of Mourre's commutator analysis is given in @cite_1 . The Floquet-Bloch theory for periodic differential operators is generalized to more general covering graphs in @cite_39 , @cite_53 . Random walk is often used to study the structure of the graph, see e.g. @cite_23 and @cite_13 . Determination of spectra, spectral gap, and (non) existence of eigenvalues are basic issues for periodic, or more generally, covering graphs, and many works are now presented, e.g. @cite_45 , @cite_41 , @cite_30 , @cite_24 , @cite_26 , @cite_56 .
@cite_30 @cite_26 @cite_41 @cite_53 @cite_1 @cite_39 @cite_24 @cite_19 @cite_56 @cite_45 @cite_23 @cite_16 @cite_13
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Abstract We consider Schrodinger operators with periodic potentials on periodic discrete graphs. The spectrum of the Schrodinger operator consists of an absolutely continuous part (a union of a finite number of non-degenerated bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We obtain estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graph and show that they become identities for some class of graphs. Moreover, we obtain stability estimates and show the existence and positions of large number of flat bands for specific graphs. The proof is based on the Floquet theory and the precise representation of fiber Schrodinger operators, constructed in the paper. In this paper we study the spectral structure of the discrete Laplacian on an infinite graph. We show that, for a finite graph including a certain kind of a family of cycles, the spectrum of the Laplacian on its homology universal covering graph has band structure and no eigenvalues; furthermore it is purely absolutely continuous. Interesting examples that illustrate our theorems are also exhibited. We consider the problem of designing spectral graph filters for the construction of dictionaries of atoms that can be used to efficiently represent signals residing on weighted graphs. While the filters used in previous spectral graph wavelet constructions are only adapted to the length of the spectrum, the filters proposed in this paper are adapted to the distribution of graph Laplacian eigenvalues, and therefore lead to atoms with better discriminatory power. Our approach is to first characterize a family of systems of uniformly translated kernels in the graph spectral domain that give rise to tight frames of atoms generated via generalized translation on the graph. We then warp the uniform translates with a function that approximates the cumulative spectral density function of the graph Laplacian eigenvalues. We use this approach to construct computationally efficient, spectrum-adapted, tight vertex-frequency and graph wavelet frames. We give numerous examples of the resulting spectrum-adapted graph filters, and also present an illustrative example of vertex-frequency analysis using the proposed construction. We use the generalization of the Laplacian matrix to hypergraphs to obtain several spectral-like results on hypergraphs. For instance, we obtain upper bounds on the eccentricity and the excess of any vertex of hypergraphs. We extend to the case of hypergraphs the concepts of walk regularity and spectral regularity, showing that all walk-regular hypergraphs are spectrally-regular. Finally, we obtain an upper bound on the mean distance of walk-regular hypergraphs that involves all the Laplacian spectrum.