Some problems in inverse scattering theory
The spectrum of the Laplacian on manifolds with cylindrical ends consists of continuous spectrum of locally finite multiplicity and embedded eigenvalues. We prove a Weyl-type asymptotic formula for the sum of the number of embedded eigenvalues and the scattering phase. In particular, we obtain the optimal upper bound on the number of embedded eigenvalues less than or equal to , where is the dimension of the manifold.
We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of , and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we...
We discuss spectral and scattering theory of the discrete laplacian limited to a half-space. The interesting properties of such operators stem from the imposed boundary condition and are related to certain phenomena in surface physics.
On s'intéresse dans cet article, a la stabilisation de l'équation des ondes dans un domaine extérieur avec condition de Dirichlet...
We prove an estimate of the kind , where , is the scattering amplitude related to the compactly supported potential at a fixed energy level const., , and is a suitably defined norm.
We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation in a time-dependent potential and we show the existence of the wave operator in Schatten spaces.