185 research outputs found

    Inverse problems, trace formulae for discrete Schr\"odinger operators

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    We study discrete Schroedinger operators with compactly supported potentials on the square lattice. 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 the spectral shift function and the potential.Comment: Ann. Henri Poincar\'e, 201

    On the S-matrix for three body Schrodinger Operators

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    Forward and inverse scattering on manifolds with asymptotically cylindrical ends

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    We study an inverse problem for a non-compact Riemannian manifold whose ends have the following properties : On each end, the Riemannian metric is assumed to be a short-range perturbation of the metric of the form (dy)2+h(x,dx)(dy)^2 + h(x,dx), h(x,dx)h(x,dx) being the metric of some compact manifold of codimension 1. Moreover one end is exactly cylindrical, i.e. the metric is equal to (dy)2+h(x,dx)(dy)^2 + h(x,dx). Given two such manifolds having the same scattering matrix on that exactly cylindrical end for all energy, we show that these two manifolds are isometric

    Spectral theory and inverse problem on asymptotically hyperbolic orbifolds

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    We consider an inverse problem associated with nn-dimensional asymptotically hyperbolic orbifolds (n2)(n \geq 2) having a finite number of cusps and regular ends. By observing solutions of the Helmholtz equation at the cusp, we introduce a generalized SS-matrix, and then show that it determines the manifolds with its Riemannian metric and the orbifold structure
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