Recovery of singularities in inverse scattering
- Autores: Cristobal Jacobo Meroño Moreno
- Directores de la Tesis: Alberto Ruiz González (dir. tes.), J. M. Barceló Valcárcel (dir. tes.)
- Lectura: En la Universidad Autónoma de Madrid ( España ) en 2008
- Idioma: inglés
- Número de páginas: 104
- Tribunal Calificador de la Tesis: Ana Vargas Rey (presid.), Pedro Pérez Caro (secret.), David Dos Santos Ferreira (voc.)
- Programa de doctorado: Programa de Doctorado en Matemáticas por la Universidad Autónoma de Madrid
- Materias:
- Enlaces
- Tesis en acceso abierto en: Biblos-e Archivo
- Resumen
The central problem in inverse scattering for the Schrödinger equation is to recover an unknown potential q(x), from the scattering data, the so called far field pattern or scattering amplitude. We consider formally well posed inverse problems by restricting the domain of the scattering data to depend only on n parameters. This makes sense from the point of view of applications, where is natural to reduce the number of measurements as much as possible. There are different ways to do this.
One of the most widely studied is the backscattering problem. In this case, as the name suggest, only the waves scattered in the opposite direction of the incident wave (the echoes) are taken into account. This is the main problem studied in this thesis, but we will also treat the fixed angle scattering problem, where the waves come instead from a fixed direction.
In these scattering problems, the usual procedure is to construct the Born approximation of the potential, which we denote by qB(x) in the case of backscattering. The Born approximation is essentially the Fourier inverse transform of the restricted scattering data. In a certain sense, it is a linear approximation to the inverse problem and it is widely used in applications. In fact, as the name suggest, the Born approximation is a good approximation for potentials satisfying certain smallness conditions, in the sense that the difference q-qB is also small in appropriate function spaces.
From a mathematical point of view an important question that is still not completely answered is to establish how much information does the Born approximation contain about the actual potential q(x), and if it is possible to recover q(x) completely from the knowledge of qB(x). Motivated by the use of the Born approximation in applications, and in search of partial recovery results, another approach is to ask how much and what kind of information about q(x) can be obtained just by looking at qB(x), that is, in a very immediate way.
In this sense, it has been shown that the Born must contain the leading singularities of q(x). Since then, this approach has received great amount of attention in different scattering problems.
The main objective of this work is to quantify as exactly as possible how much more regular than q(x) can q-qB be in general, depending on the dimension n, and the a priori regularity of the potential q(x) measured in the Sobolev scale. We address the same question in fixed angle scattering. The potentials considered can be complex valued and we don't assume any smallness condition on them.
The main results obtained in this thesis can be divided in three groups. In the first place we study the recovery of singularities in backscattering in the Sobolev scale and in general dimension, improving most of the previous known results. Moreover, for the first time we show that there is a necessary condition which limits the amount of singularities that can be recovered from the Born approximation.
In the second place, we try to obtain optimal results of recovery of singularities (optimal according to the previous results). We follow two approaches: one imposing extra conditions on the potentials, and the other choosing a weaker scale to measure the regularity of q-qB.
In the third place, we extend the techniques developed in the first part of the thesis to study the fixed angle scattering problem, which yields analogous results of recovery of singularities.
