The main question that has been answered in this research project is: what happens to the singularities of the S-matrix when the interaction potential becomes complex? In other words, we looked at the resonance spectral points and traced their movements on the complex momentum plane, when the imaginary part of the potential is gradually increased from zero to a certain reasonable value. This question is important in many fields of research where optical potentials are used. An optical potential can effectively take into account the loss of the beam of particles into all the reaction channels that are formally ignored. Such a simplified approach is widely used in nuclear and atomic physics. In order to study the movement of the spectral points, we used a simple potential and the Jost function method that allowed us to easily locate the spectral points as complex zeros of the Jost function. / Dissertation (MSc)--University of Pretoria, 2016. / Physics / MSc / Unrestricted

This thesis consists of a summary of four papers dealing with resonances of Dirac operators on Euclidean 3-space. In Paper I we show that the Complex Absorbing Potential (CAP) method is valid in the semiclassical limit for resonances sufficiently close to the real line if the potential is smooth and compactly supported. In Paper II  we continue the investigations initiated in Paper I but here we study clouds of resonances close to the real line and show that in some sense the CAP method remains valid also for multiple resonances. In Paper III we study perturbations of Dirac operators with smooth decaying scalar potentials  and show that these possess many resonances near certain points related to the maximum and the minimum of the potential. In Paper IV we show a trace formula of Poisson type for Dirac operators having compactly supported potentials which is related to resonances. The techniques mainly stem from complex function theory and scattering theory.

Depuis la fin des années 80 et l'avènement de la femto-chimie nous pouvons sonder la dynamique nucléaire à l’œuvre au cours de réactions chimiques à l'échelle de la femtoseconde. Plus récemment, la production d'impulsions lasers attosecondes isolées permet d'atteindre une résolution temporelle plus grande encore. Par elle, il devient possible de sonder la dynamique d'origine purement électronique induite par photo-excitation, et notamment photo-ionisation. Dans ce contexte, avec le développement des techniques de spectroscopie résolue en temps, il est important de disposer d'approches théoriques fiables aidant à l'appréhension de résultats toujours plus nombreux dans ce domaine. La tâche et néanmoins rendue difficile par le caractère profondément multi-électronique des processus en jeu. Traiter de tels effets précisément requiert une grande puissance de calcul, ce qui a limité les études disponibles aujourd'hui à de petits systèmes. Au cours de cette thèse, j'ai tenté d'expliquer les résultats d'une expérience de type "pompe-sonde" (UVX-IR) sur molécule de Caféine menée par une équipe de collaborateurs à l'Institut lumière matière. J'ai utilisé pour cela une méthode basée sur la théorie de la fonctionnelle de la densité dépendante du temps, la TD-DFTB dont le coût numérique réduit par rapport à cette dernière permet des calculs sur de gros systèmes en temps raisonnable. J'y présente une étude du paysage énergétique de la Caféine ainsi que le résultat de 2 approches distinctes pour simuler l'ionisation de ce composé. La première, l'approximation de l'ionisation soudaine cosiste à retirer "à la main" un électron à l'une des orbitales Kohn-Sham occupées du système neutre et ne tient pas compte du champ laser. La seconde à recours à un potentiel imaginaire (ou CAP - Complex Absorbing Potential) pour simuler la perte d'electrons, et tiens explicitement compte du champ laser / Since the advent of femtochemistry, at the end of 1980's, we are able to probe the nuclear dynamics underlying chemical reactions down to the scale of a femtosecond. More recently, the production of isolated attosecond pulses allows to reach an even bigger temporal resolution. It is now possible to probe the ultrafast electronic dynamics following a photo-excitation. In this context, with the developpement of time-resolved spectroscopy techniques, it is important to have reliable theorectical approaches in order to apprehend the increasing number of results in this field. This task is made difficult by the intrinsic multi-electronic nature processes at play. The precise treatment of such effects requires a considerable computing power, and have thus limited the availables studies to relatively small systems. In this thesis, I tried to explain the outcome of a "pump-probe" (XUV-IR) experiment on Caffeine molecule realized by our collaborators at the Insitut Lumière Matière. To do so, I used a method based on density functional theory, the TD-DFTB, which lower numerical cost with respect to TD-DFT allows calculation on bigger compounds. I present in the document a study of the energetical landscape of Caffeine, and 2 approaches to simulate ionization. The first one, the so called sudden-ionization approximation consist to retrieve "by hand" an electron from the occupied Kohn-Sham orbitals of the neutral system without taking the laser field into account. The other one is based on the introduction of a complex absorbing potential (CAP) to account for electron loss and take explicitely the laser field into account.