Alberto Ruiz

Partial Recovery of a Potential from Backscattering Data

ABSTRACT We prove that the main singularities,measured in the scale of Sobolev spaces,of the potential q in the Schrödinger Hamiltonian −Δ+q,in dimensions n=2,3,are contained in the Born approximation for backscattering data.

UNIQUE CONTINUATION FOR SCHRODINGER OPERATORS WITH POTENTIAL IN MORREY SPACES

Let us consider in a domain O of Rn solutions of the differential inequality |?u(x)| = V(x)|u(x)|, x I O, where V is a non smooth, positive potential. We are interested in global unique continuation properties. That means that u must be identically zero on O if it vanishes on an open subset of O.

RECOVERY OF THE SINGULARITIES OF A POTENTIAL FROM FIXED ANGLE SCATTERING DATA

We study the inverse scattering problem for Schroedinger equation. We prove that for non-smooth potential the main singularities of the potential are contained in the Born approximation which can be obtained from measurement of the scattering amplitude in a single outgoing direction. We measure singularities in the scale of Sobolev spaces.

Corrigenda to "Unique continuation for Schrödinger operators" and a remark on intepolation of Morrey spaces

The purpose of this note is twofold. First it is a corrigenda of our paper \cite{RV1}. And secondly we make some remarks concerning the interpolation properties of Morrey spaces

Weak Dispersive Estimates for Schrödinger Equations with Long Range Potentials

We prove local smoothing estimates for the Schrödinger initial value problem with data in the energy space L 2(ℝ d ), d ≥ 2 and a general class of potentials. In the repulsive setting we have to assume just a power like decay (1 + |x|)−γ for some γ > 0. Also attractive perturbations are considered. The estimates hold for all time and as a consequence a weak dispersion of the solution is obtained. The proofs are based on similar estimates for the corresponding stationary Helmholtz equation and Kato H-smooth theory.

A priori estimates for the Helmholtz equation with electromagnetic potentials in exterior domains

We study the Helmholtz equation with electromagnetic-type perturbations, in the exterior of a domain, in dimension

Strichartz inequalities with weights in Morrey-Campanato classes

n\geq3

Resolvent and Strichartz estimates for elastic wave equations

. We prove, by multiplier techniques in the sense of Morawetz, a family of a priori estimates from which the limiting absorption principle follows. Moreover, we give some standard applications to the absence of embedded eigenvalues and zero-resonances, under explicit conditions on the potentials.

A Born Approximation for Live Loads in Navier Elasticity

We prove some weighted refinements of the classical Strichartz inequalities for initial data in the Sobolev spaces Ḣs(ℝn). We control the weightedL2-norm of the solution of the free Schrödinger equation whenever the weight is in a Morrey-Campanato type space adapted to that equation. Our partial positive results are complemented by some necessary conditions based on estimates for certain particular solutions of the free Schrödinger equation.

Stability estimates for a magnetic Schrödinger operator with partial data

Abstract We show a simple argument to prove resolvent estimates for Lame operators of elasticity, with constant coefficients, and Strichartz estimates for the associated elastic wave equation. The argument is based on the Helmholtz decomposition for vector fields and some elliptic regularity result in weighted L 2 -spaces. By standard perturbation arguments, we can include small critical 0-order potentials in the main results.