2,836 research outputs found
Inverse problems, trace formulae for discrete Schr\"odinger operators
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
Forward and inverse scattering on manifolds with asymptotically cylindrical ends
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 ,
being the metric of some compact manifold of codimension 1. Moreover
one end is exactly cylindrical, i.e. the metric is equal to .
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
We consider an inverse problem associated with -dimensional asymptotically
hyperbolic orbifolds having a finite number of cusps and regular
ends. By observing solutions of the Helmholtz equation at the cusp, we
introduce a generalized -matrix, and then show that it determines the
manifolds with its Riemannian metric and the orbifold structure
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