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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Let G be an infinite d-regular graph and L(G) its line graph. We consider discrete Laplacians on G and L(G), and show the exact relation between the spectrum of -AG and that of -AL(G). Our method is also applicable to (di, d2)-semiregular graphs, subdivision graphs and para-line graphs. 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. Abstract We develop in this paper the Mourre theory for an abstract class of fibered self-adjoint operators which we call analytically fibered operators. We construct a conjugate operator for which we prove that a Mourre estimate holds. Examples of analytically fibered operators are given and, finally, perturbations of such operators are considered. 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. Abstract In this paper, we examine covering graphs that are obtained from the d -dimensional integer lattice by adding pendant edges. In the case of d = 1 , we show that the Laplacian on the graph has a spectral gap and establish a necessary and sufficient condition under which the Laplacian has no eigenvalues. In the case of d = 2 , we show that there exists an arrangement of the pendant edges such that the Laplacian has no spectral gap. The difference Laplacian on a square lattice in Rn has been stud- ied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference oper- ator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient. The random walks we consider are defined as follows. Let K be a connected graph (i.e. a one dimensional simplicial complex). For a vertex x E K, let m(x) denote the number of edges emanating from x. The probability that a particle moves from x to another vertex y E K is l m(x) if x and y are connected by an edge and it is zero otherwise. As observed by Courant, Friedrichs and Lewy (CFL) for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the Laplacian Eigenvalues and the Laplacian of a graph Isoperimetric problems Diameters and eigenvalues Paths, flows, and routing Eigenvalues and quasi-randomness Expanders and explicit constructions Eigenvalues of symmetrical graphs Eigenvalues of subgraphs with boundary conditions Harnack inequalities Heat kernels Sobolev inequalities Advanced techniques for random walks on graphs Bibliography Index. Ideas cultivated in spectral geometry are applied to obtain an asymptotic property of a reversible random walk on an infinite graph satisfying a certain periodic condition. In the course of our argument, we employ perturbation theory for the maximal eigenvalues of twisted transition operator. As a result, an asymptotic of the probabilityp(n, x, y) that a particle starting atxreachesyat timenasngoes to infinity is established.