Related papers: Forward and inverse scattering on manifolds with a…
The problems we address in this paper are the spectral theory and the inverse problems associated with Laplacians on non-compact Riemannian manifolds and more general manifolds admitting conic singularities. In particular, we study the…
We give a brief survey for the recent development of inverse scattering theory on non-compact Riemannian manifolds. The main theme is the reconstruction of the manifold and the metric from the scattering matrix.
We consider a fixed angle inverse scattering problem in the presence of a known Riemannian metric. First, assuming a no caustics condition, we study the direct problem by utilizing the progressing wave expansion. Under a symmetry assumption…
Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, $g.$ A model form is established for such metrics close to the boundary. It is shown that the scattering matrix at energy…
We study inverse scattering for $\Delta_g+V$ on $(X,g)$ a conformally compact manifold with metric $g,$ with variable sectional curvature $-\alf^2(y)$ at the boundary and $V\in C^\infty(X)$ not vanishing at the boundary. We prove that the…
In this paper, we study an inverse scattering problem at fixed energy on three-dimensional asymptotically hyperbolic St{\"a}ckel manifolds having the topology of toric cylinders and satisfying the Robertson condition. On these manifolds the…
Given a smooth non-trapping compact manifold with strictly con- vex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. This data consist of the exit directions…
We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = \Gamma\backslash{\bf H}^2$ associated with…
We prove that the L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L^2 metric is a weak Riemannian metric, this fact does not…
We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in…
The problem of recovering the asymptotics of a short range perturbation of the Euclidean metric on R^n from fixed energy scattering data is studied. It is shown that if two such metrics, g1, g2, have scattering data at some fixed energy…
We show that an inverse scattering problem for a semilinear wave equation can be solved on a manifold having an asymptotically Minkowskian infinity, that is, scattering functionals determine the topology, differentiable structure, and the…
We prove that the scattering matrix at all energies restricted to an open subset of the boundary determines an asymptotically hyperbolic manifold modulo isometries that are equal to the identity on the open subset where the scattering…
Scattering theory for p-forms on manifolds with cylindrical ends has a direct interpretation in terms of cohomology. Using the Hodge isomorphism,the scattering matrix at low energy may be regarded as operator on the cohomology of the…
We consider a strongly damped wave equation on compact manifolds, both with and without boundaries, and formulate the corresponding inverse problems. For closed manifolds, we prove that the metric can be uniquely determined, up to an…
We analyze the resolvent and define the scattering matrix for asymptotically hyperbolic manifolds with metrics which have a polyhomogeneous expansion near the boundary, and also prove that there is always an essential singularity of the…
We study scattering and inverse scattering theories for asymptotically complex hyperbolic manifolds. We show the existence of the scattering operator as a meromorphic family of operators in the Heisenberg calculus on the boundary, which is…
The inverse scattering transform for the focusing nonlinear Schrodinger equation is presented for a general class of initial conditions whose asymptotic behavior at infinity consists of counterpropagating waves. The formulation takes into…
The quantum-mechanical scattering on a compact Riemannian manifold with semi-axes attached to it (hedgehog-shaped manifold) is considered. The complete description of the spectral structure of Schroedinger operators on such a manifold is…
For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result…