Search results for "35J10"
showing 4 items of 4 documents
The Calderón problem for the fractional Schrödinger equation
2020
We show global uniqueness in an inverse problem for the fractional Schr\"odinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where the measurements are taken in arbitrary open, possibly disjoint, subsets of the exterior. The results apply in any dimension $\geq 2$ and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically different from the traditional Calder\'on problem.
Adiabatic evolution and shape resonances
2017
Motivated by a problem of one mode approximation for a non-linear evolution with charge accumulation in potential wells, we consider a general linear adiabatic evolution problem for a semi-classical Schrödinger operator with a time dependent potential with a well in an island. In particular, we show that we can choose the adiabatic parameter ε \varepsilon with ln ε ≍ − 1 / h \ln \varepsilon \asymp -1/h , where h h denotes the semi-classical parameter, and get adiabatic approximations of exact solutions over a time interval of length ε − N \varepsilon ^{-N} with an error O ( ε N ) {\mathcal O}(\varepsilon ^N) . Here N > 0 N>0 is arbitrary. \center Résumé \endcenter Motivés par un pro…
The Calderón problem with partial data on manifolds and applications
2013
We consider Calderon's inverse problem with partial data in dimensions $n \geq 3$. If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility of a broken geodesic ray transform. In Euclidean space, sets satisfying the flatness condition include parts of cylindrical sets, conical sets, and surfaces of revolution. We prove local uniqueness in the Calderon problem with partial data in admissible geometries, and global uniqueness under an additional concavity assumption. This work unifies two earlier approaches to this problem (\cite{KSU} and \cite{I}) and extends both. The proofs are based on impr…
Resonances over a potential well in an island
2020
In this paper we study the distribution of scattering resonances for a multidimensional semi-classical Schr\"odinger operator, associated to a potential well in an island at energies close to the maximal one that limits the separation of the well and the surrounding sea.