Inverse scattering results for manifolds hyperbolic near infinity

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 dimension three. In dimensions four or higher we prove topological finiteness theorems under the negative curvature assumption.Comment: 25 pages. v3: Minor corrections, references adde

CR-Invariants and the Scattering Operator for Complex Manifolds with Boundary

The purpose of this paper is to describe certain CR-covariant differential operators on a strictly pseudoconvex CR manifold $M$ as residues of the scattering operator for the Laplacian on an ambient complex K\"{a}hler manifold $X$ having $M$ as a `CR-infinity.' We also characterize the CR $Q$-curvature in terms of the scattering operator. Our results parallel earlier results of Graham and Zworski \cite{GZ:2003}, who showed that if $X$ is an asymptotically hyperbolic manifold carrying a Poincar\'{e}-Einstein metric, the $Q$-curvature and certain conformally covariant differential operators on the `conformal infinity' $M$ of $X$ can be recovered from the scattering operator on $X$. The results in this paper were announced in \cite{HPT:2006}.Comment: 32 page

Isospectral Sets for Fourth-Order Ordinary Differential Operators

Let L(p)u = D4u - (p1u’)’ + p2u be a fourth-order differential operator acting on L2[0; 1] with p ≡ (p1; p2) belonging to L2ℝ[0, 1] x L2ℝ[0, 1] and boundary conditions u(0) = u\u27\u27(0) = u(1) = u\u27\u27(1) = 0. We study the isospectral set of L(p) when L(p) has simple spectrum. In particular we show that for such p, the isospectral manifold is a real-analytic submanifold of L2ℝ[0, 1] x L2ℝ[0, 1] which has infinite dimension and codimension. A crucial step in the proof is to show that the gradients of the eigenvalues of L(p) with respect to p are linearly independent: we study them as solutions of a non-self-ajdoint fifth-order system, the Borg system, among whose eigenvectors are the gradients