Holmberg, Manu
(Helsingin yliopisto, 2020)
This thesis considers certain mathematical formulation of the scattering phenomena. Scattering is a common physical process, where some initial wave is disturbed, producing a scattered wave. If the direct problem is to determine the scattered wave from the knowledge of the object that causes the scattering as well and the initial wave, then the inverse problem would be to determine the object from the knowledge on how different waves scatter from it.
In this thesis we consider direct and inverse scattering problems governed by Helmholtz equation $\Delta u + k^2 \eta u = 0$ in $\mathbb{R}^d$ with $d = 3$. The positive function $\eta \in L^\infty(\mathbb{R}^d)$ is considered to be such that $\eta(x) = 1$ outside of some ball. In particular the function $\eta$ models the physical properties of the scattering object and in a certain physical setting, the function $n = +\sqrt{\eta}$ is the index of refraction. The initial motivation for this thesis was the inverse scattering problem and its uniqueness. However, for any inverse problem, one first has to understand the corresponding direct problem. In the end, the balance between treating the direct and inverse problem is left fairly even. This thesis closely follows books by Colton and Kress, and Kirsch.
The first chapter is the introduction, in which the overview of the thesis is presented and the working assumptions are made. The second chapter treats the needed preliminaries, such as compact operators, Sobolev spaces, Fredholm alternative, spherical harmonics and spherical Bessel functions. In particular these are needed in various results of chapter three, in which the direct scattering problem is considered. After motivating and defining the direct scattering problem, the main goal is to prove its well-posedness. The uniqueness of the problem is proved by two results, Rellich's lemma and unique continuation principle. The Fredholm alternative is applied to prove existence of the solution on the basis of uniqueness. Equipped with the understanding of the direct scattering problem, the inverse scattering problem can be considered in the fourth chapter. After defining the inverse scattering problem, the uniqueness of the solution is considered. The proof is contrasted to the historically important paper by Calderón considering another kind of inverse problem. The proof consists of three lemmas, from which the second and third are directly used in proving the uniqueness of the inverse problem. The uniqueness of the inverse problem can be considered as the main result of this thesis.