Fixed angle scattering: Recovery of singularities and its limitations
| Autor: | Cristóbal J. Meroño |
|---|---|
| Přispěvatelé: | UAM. Departamento de Matemáticas |
| Rok vydání: | 2018 |
| Předmět: |
Born approximation
Matemáticas 01 natural sciences Schrödinger equation Fixed angle scattering data symbols.namesake Mathematics - Analysis of PDEs Dimension (vector space) Complex potential q 35R30 35J05 35P25 42B20 81U40 FOS: Mathematics 0101 mathematics Mathematics Sobolev scale Scattering Applied Mathematics 010102 general mathematics Mathematical analysis Inverse problem 010101 applied mathematics Computational Mathematics Inverse scattering problem symbols A priori and a posteriori Gravitational singularity Dimension Analysis Analysis of PDEs (math.AP) |
| Zdroj: | Biblos-e Archivo. Repositorio Institucional de la UAM instname |
| Popis: | We prove that in dimension $n \ge 2$ the main singularities of a complex potential $q$ having a certain a priori regularity are contained in the Born approximation $q_\theta$ constructed from fixed angle scattering data. Moreover, ${q-q_\theta}$ can be up to one derivative more regular than $q$ in the Sobolev scale. In fact, this result is optimal, we construct a family of compactly supported and radial potentials for which it is not possible to have more than one derivative gain. Also, these functions show that for $n>3$, the maximum derivative gain can be very small for potentials in the Sobolev scale not having a certain a priori level of regularity which grows with the dimension. Comment: 22 pages. arXiv admin note: text overlap with arXiv:1709.00748 |
| Databáze: | OpenAIRE |
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