Antônio Sá Barreto

Inverse scattering on asymptotically hyperbolic manifolds

Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric,

Asymptotics of Solutions of the Wave Equation on de Sitter-Schwarzschild Space

g.

Recovering asymptotics of metrics from fixed energy scattering data

A model form is established for such metrics close to the boundary. It is shown that the scattering matrix at energy

Scattering and Inverse Scattering on ACH Manifolds

\zeta

EXISTENCE OF RESONANCES IN THREE DIMENSIONS

exists and is a pseudo-differential operator of order

Radiation fields, scattering, and inverse scattering on asymptotically hyperbolic manifolds

2\zeta+1 - \dim X.

On the heat trace of schrödinger operators

The symbol of the scattering matrix is then used to show that except for a countable set of energies the scattering matrix at one energy determines the diffeomorphism class of the metric modulo terms vanishing to infinite order at the boundary. An analogous result is proved for potential scattering. The total symbol is computed when the manifold is hyperbolic or is of product type modulo terms vanishing to infinite order at the boundary. The same methods are then applied to studying inverse scattering on the Schwarzschild and De Sitter-Schwarzschild models of black holes.

Radiation Fields on Asymptotically Euclidean Manifolds

Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately compactified space.

Interactions of conormal waves for fully semilinear wave equations

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 which are equal up to smoothing, then there exists a diffeomorphism ψ‘fixing infinity’ such that ψ*g1-g2 is rapidly decreasing. Given the scattering matrix at two energies, it is shown that the asymptotics of a metric and a short range potential can be determined simultaneously. These results also hold for a wide class of scattering manifolds.

Radiation fields for semilinear wave equations

Abstract 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 a contact manifold with a pseudohermitian structure. Then we define the radiation fields as in the real asymptotically hyperbolic case, and reconstruct the scattering operator from those fields. As an application we show that the manifold, including its topology and the metric, are determined up to invariants by the scattering matrix at all energies.