One recipe for mathematically formulating a relativistic quantum mechanical scattering theory utilizes a two-Hilbert space approach, denoted by $\mathcal{H}$ and $\mathcal{H}_{0}$, upon each of which a unitary representation of the Poincaré Lie group is given. Physically speaking, $\mathcal{H}$ models a complicated interacting system of particles one wishes to understand, and $\mathcal{H}_{0}$ an associated simpler (i.e., free/noninteracting) structure one uses to construct 'asymptotic boundary conditions" on so-called scattering states in $\mathcal{H}$. Simply put, $\mathcal{H}_{0}$ is an attempted idealization of $\mathcal{H}$ one hopes to realize in the large time limits $t\rightarrow\pm\infty$. The above considerations lead to the study of the existence of strong limits of operators of the form $e^{iHt}Je^{-iH_{0}t}$, where $H$ and $H_{0}$ are self-adjoint generators of the time translation subgroup of the unitary representations of the Poincaré group on $\mathcal{H}$ and $\mathcal{H}_{0}$, and $J$ is a contrived mapping from $\mathcal{H}_{0}$ into $\mathcal{H}$ that provides the internal structure of the scattering asymptotes. The existence of said limits in the context of Euclidean quantum theories (satisfying precepts known as the Osterwalder-Schrader axioms) depends on the choice of $J$ and leads to a marvelous connection between this formalism and a beautiful area of classical mathematical analysis known as the Stieltjes moment problem, which concerns the relationship between numerical sequences $\{\mu_{n}\}_{n=0}^{\infty}$ and the existence/uniqueness of measures $\alpha(x)$ on the half-line satisfying \begin{equation*} \mu_{n}=\int_{0}^{\infty}x^{n}d\alpha(x). \end{equation*}

We consider scattering in $R^n$, $nge 2$, described by the Schr"odinger operator $P(h)=-h^2Delta+V$, where $V$ is a short-range potential. With the aid of Maslov theory, we give a geometrical formula for the semiclassical asymptotics as $hto 0$ of the scattering amplitude $f(omega_-,omega_+;lambda,h)$ $omega_+neqomega_-$) which remains valid in the presence of focal points at infinity (caustics). Crucial for this analysis are precise estimates on the asymptotics of the classical phase trajectories and the relationship between caustics in euclidean phase space and caustics at infinity. / Wir betrachten Streuung in $R^n$, $nge 2$, beschrieben durch den Schr"odinger operator $P(h)=-h^2Delta+V$, wo $V$ ein kurzreichweitiges Potential ist. Mit Hilfe von Maslov Theorie erhalten wir eine geometrische Formel fuer die semiklassische Asymptotik ($hto 0$) der Streuamplitude $f(omega_-,omega_+;lambda,h)$ ($omega_+neqomega_-$) welche auch bei Vorhandensein von Fokalpunkten bei Unendlich (Kaustiken) gueltig bleibt.

A rigorous formalism for solving the Coulomb scattering problem is presented in this thesis. The approach is based on splitting the interaction potential into a finite-range part and a long-range tail part. In this representation the scattering problem can be reformulated to one which is suitable for applying exterior complex scaling. The scaled problem has zero boundary conditions at infinity and can be implemented numerically for finding scattering amplitudes. The systems under consideration may consist of two or three charged particles. The technique presented in this thesis is first developed for the case of a two body single channel Coulomb scattering problem. The method is mathematically validated for the partial wave formulation of the scattering problem. Integral and local representations for the partial wave scattering amplitudes have been derived. The partial wave results are summed up to obtain the scattering amplitude for the three dimensional scattering problem. The approach is generalized to allow the two body multichannel scattering problem to be solved. The theoretical results are illustrated with numerical calculations for a number of models. Finally, the potential splitting technique is further developed and validated for the three body Coulomb scattering problem. It is shown that only a part of the total interaction potential should be split to obtain the inhomogeneous equation required such that the method of exterior complex scaling can be applied. The final six-dimensional equation is reduced to a system of three dimensional equations using the full angular momentum representation. Such a system can be numerically implemented using the existing full angular momentum complex exterior scaling code (FAMCES). The code has been updated to solve the three body scattering problem. / At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Submitted. Paper 5: Manuscript.

New methods of biological analyte sensing are needed for development of miniature biosensors that are highly sensitive and require minimal sample preparation. One novel technique employs optical resonances known as Whispering Gallery Modes (WGMs). These modes arise from total internal reflection of light at the internal surface of a high index microsphere within a low index medium and produce an evanescent field that extends into the surrounding medium. The WGMs produce multiple narrow spectral peaks that shift position with variations in the local index of refraction sampled by the evanescent tail of the WGMs. To excite these WGMs, we embed quantum dots (QDs) in the periphery of polystyrene microspheres to serve as local light sources. By coupling emission from the QDs to the WGMs, the sensors can be excited and interrogated remotely and, by monitoring the shift of multiple resonance modes, may provide higher sensitivity and accuracy compared with similar techniques. The high refractometric sensitivity of the WGMs offers potential for trace detection of molecules adsorbed onto or bound to the microsphere sensor elements. The sensitivity of these sensors is demonstrated by monitoring the wavelength shift of multiple resonant modes as bulk index of refraction is changed. The potential for targeted biosensing is explored through addition of a protein that adsorbs to the microsphere surface, thrombin. Microsensor response in all cases demonstrated increased sensitivity over theoretical predictions. Models based on Mie theory and continuity of the radial functions across the sphere-media interface were used to model the location, Q-factor, and sensitivity of the WGMs in microspheres by considering the embedded QDs as a high index outer layer. This model was used, along with estimates of the QD-layer index and penetration depth, to relate the locations and sensitivities of the modes to our experimental results with strong agreement between the two. In all, these microspheres demonstrate feasibility for use as remote microsensors with sensitivities rivaling current techniques.

A particularly interesting class of modern materials is ferroic ceramics. Their characteristic order parameter is a result of quantum chemistry taking place on a sub-Å length scale and long-range couplings, e.g. mediated by electrostatic or stress fields. Furthermore, the particular subclass of multiferroics possesses more than one order parameter and exhibits an intriguing coupling between them, which is interesting both from the fundamental physics point of view as well as from a technological vantage point. While on a more fundamental level it is desirable to elucidate the physical details of the coupling mechanism, this knowledge could subsequently lead to new and technologically interesting multiferroic materials, which overcome their current drawback that only one of the multiple order parameters is appreciably large while the others stay small. Due to the short and long range nature of the driving forces, one challenge for thoroughly understanding ferroic ceramics is the characterization of material properties within a large interval of length scales from several tens of µm to sub-Å. To that end, it is useful to exploit that all order parameters can be described as macroscopic fields, e.g. electric polarization or strain, which, in turn, can be either directly or indirectly probed with an electron beam such as used in Transmission Electron Microscopy (TEM). Consequently, TEM is excellently suited for investigating ferroic materials, i.e., state-of-the-art instruments facilitate aberration corrected imaging within a large magnification interval covering the length scales of interest, in particular the atomic regime. A general drawback of conventional TEM techniques is the loss of phase information originally contained in the scattered electron wave introduced by recording only the electron density. Electron Holography is an advanced TEM technique that facilitates the complete evaluation of the complex electron wave, which, in combination with the manifold possibilities of TEM, provides rather straightforward access to static electromagnetic fields within the ceramic. Nevertheless, quantification of order parameters such as the electric polarization or minute details in electromagnetic fields still require to correlate the experimentally gained observations to physical models, which combine the details of the microscopic imaging process, the electron-specimen scattering, and solid state physics of the specimen. The goal of this work is to investigate and advance the limits of Electron Holography as a truly quantitative TEM technique and apply the findings in, e.g., the investigation of ferroic ceramics. In the light of the previously mentioned difficulties, the problem has to be tackled from different directions: Firstly, the whole holographic imaging process is reviewed and extended, if necessary, in order to provide quantitative measures for systematic and statistical errors inherent to reconstructed waves. In the course of that process, two previously not recognized holography-specific aberrations are identified, firstly, a resolution limiting spatial envelope and secondly, a spatial distortion to the reconstructed wave. Furthermore, several correction strategies have been developed, in order to correct the aforementioned two and other well-known disturbances, e.g. Fresnel fringes from the biprism filament. The previous holographic noise model has been extended to incorporate the important contribution from the detector and consequently to provide realistic statistic error bars of the holographically reconstructed amplitude and phase. Secondly, an investigation of the electron-specimen scattering process itself is conducted, leading to a density matrix description of the holographic measurement. The general laws of quantum electrodynamics provide the framework of that description. Relativistic phenomena such as retardation of electromagnetic fields exchanged between beam electron and specimen and spin-orbit coupling of the beam electron are quantified, where the latter is found to be negligible within TEM. The decoherence of the electron wave by statistical coupling to the thermally moving crystal lattice of ceramics is treated by a newly developed algorithm facilitating in particular the accurate quantification of elastic scattering on heavy elements. Inelastic excitations in the ceramic, e.g. bulk plasmons or core electrons, are treated in combination with elastic scattering to identify their role in the holographic reconstruction process and to develop methods for an accurate calculation. A new scattering algorithm combining elastic and inelastic scattering is developed and applied to predict peculiar scattering contrasts of dipole transitions and to discuss the long-standing problem of contrast mismatch between scattering simulations and conventional imaging. To provide a user-friendly and continuing use of the findings, a software package SEMI (Simulation of Electron Microscopy Imaging) has been written, which facilitates the simulation of elastic and inelastic scattering processes and the subsequent imaging within different approximations, incorporating the newly developed algorithms. Thirdly, Density Function Theory (DFT) solid state calculations have been employed to identify and quantify structural modifications and characteristic electromagnetic fields, such as occurring at domain boundaries, within typical ferroic ceramics like BaTiO3 or BiFeO3, and concomitantly provide models correlating observables of the (holographic) experiment to characteristics of the materials, e.g. the order parameters. This is particularly important when static electromagnetic fields provide no direct information about the order parameter, e.g. the electric polarization, i.e., it is possible to correlate the measurable atomic positions to the electric polarization within linear response theory. A software package ATA (AuTomated Atomic contrast fitting) has been developed facilitating an automated fitting of atomic positions and a subsequent determination of local polarization. In a fourth step, electron holographic experiments analyzed with the help of the revised imaging process in combination with the knowledge gained from scattering theory are used as an input to the models established from solid state physics to yield quantitative information about bulk ferroelectric materials such as BaTiO3 and PbTiO3 and more complicated configurations such as domain walls in BiFeO3 and KnbO3. It is found that particular atomic shifts characteristic for ferroelectrics provide the most reliable quantitative information about the polarization down to nm length scales, whereas minute wave modification due to characteristic electron distributions within the ceramic are currently insufficiently quantitatively interpretable within Electron Holography. The linear response program, correlating atomic positions to ferroelectric polarization with the help of ab-initio calculated Born effective charges, has been successfully applied to determine finite size effects, screening layer widths and polarization charges in non-ferroelectric/ferroelectric layered systems. Finally, a special section considers the evaluation of 3D electromagnetic fields by Electron Holographic Tomography, which provides the means to characterize even more complex 3D domain wall configurations. As the capabilities of the technique are still limited by holographic reconstruction errors and particular tomographic issues such as incomplete projection data, the main focus of that section is put on the characterization and improvement of the tomographic reconstruction process. A Singular Value based reconstruction method is developed, which facilitates a quantification and control of the tomographic reconstruction error. Furthermore, vector field reconstruction is extended in order to treat magnetic vector fields leaking out from the reconstruction volume. / Ferroische Keramiken bilden eine besonders interessante Klasse moderner funktionaler Werkstoffe. Ihr charakteristischer Ordnungsparameter ist das Ergebnis quantenchemischer Prozesse innerhalb einer sub- Å Längenskala und spezifischer langreichweitiger Kopplungen, welche beispielsweise durch elektromagnetische oder Spannungsfelder vermittelt werden. Des Weiteren besitzt die besondere Unterklasse der Multiferroika mehr als einen Ordnungsparameter und zeigt eine faszinierende Kopplung zwischen ihnen, was sowohl vom Standpunkt physikalischer Grundlagenforschung als auch aus technologischer Sicht von Interesse ist. Während es vom fundamentalen Standpunkt erstrebenswert ist, die physikalischen Details des Kopplungsmechanismus aufzuklären, könnte in der Folge dieses Wissen zu neuen und technologisch interessanten multiferroischen Materialien führen, welche den derzeit bestehenden Nachteil, dass nur ein Ordnungsparameter genügend groß ist, während die jeweils anderen klein bleiben, hinter sich lassen. Aufgrund der kurz- und langreichweitigen Natur der Antriebskräfte besteht eine Herausforderung für das umfassende Verständnis ferroischer Keramiken aus der Charakterisierung von Materialeigenschaften innerhalb eines breiten Intervalls von Längenskalen, welches von einigen 10 µm bis unterhalb eines Å reicht. Um dieses Ziel zu erreichen ist es zweckmäßig auszunutzen, dass alle Ordnungsparameter als makroskopische, beispielsweise elektrostatische oder Verzerrungs-, Felder beschrieben werden können, welche wiederum direkt oder indirekt mit einem Elektronenstrahl, wie er im Transmissionselektronenmikrokop (TEM) zur Anwendung kommt, gemessen werden können. Folglich ist die Transmissionselektronenmikroskopie hervorragend geeignet um ferroische Materialien zu untersuchen, das heißt, modernste Geräte ermöglichen aberrationskorrigierte Aufnahmen innerhalb eines großen Vergrößerungsbereiches, welche die interessanten Längenskalen und insbesondere den atomaren Bereich abdecken. Ein allgemeiner Nachteil der konventionellen TEM Techniken ist der Verlust der Phaseninformationen, welche ursprünglich in der Elektronenwelle vorhanden sind und durch die Aufzeichnung der Elektronenintensität zerstört werden. Elektronenholographie ist eine weiterentwickelte TEM Technik, welche die vollständige Auswertung der komplexen Elektronenwelle ermöglicht, was wiederum in Verbindung mit den vielfältigen Möglichkeiten der TEM einen vergleichsweise direkten Zugang zu elektromagnetischen Feldern in der Keramik ermöglicht. Nichtsdestotrotz erfordert die Quantifizierung von Ordnungsparametern, wie der elektrische Polarisierung, oder von kleinsten Details elektromagnetischer Felder die Korrelation experimenteller Daten mit physikalischen Modellen, welche die Details des mikroskopischen Bildgebungsprozesses mit der Elektronen-Objekt Streuung und der Festkörperphysik des Objektes kombinieren. Das Ziel dieser Arbeit besteht aus der Untersuchung und Erweiterung der Möglichkeiten von Elektronenholographie als quantitative TEM Messmethode und der Anwendung dieser Ergebnisse bei der Untersuchung ferroischer Keramiken. Im Lichte der eben erwähnten Schwierigkeiten muss das Problem von verschiedenen Richtungen bearbeitet werden: Erstens wird der komplette holographische Bildgebungsprozess mit dem Ziel einer quantitativen Bewertung systematischer und statistischer Fehler der rekonstruierten Welle analysiert und gegebenenfalls erweitert. Im diesem Zuge wurden zwei bisher nicht erkannte holographiespezifische Fehler identifiziert, erstens eine auflösungsbegrenzende räumliche Enveloppe und zweitens eine räumliche Verzerrung der rekonstruierten Welle. Außerdem wurden verschiedene Korrekturmöglichkeiten entwickelt, um die zwei eben genannten und andere wohlbekannte Störungen, wie zum Beispiel die Fresnelstreifen des Biprismafadens, zu korrigieren. Das bisherige holographische Rauschmodel wurde erweitert um den beträchtlichen Einfluss des Detektors zu berücksichtigen und damit realistische Fehlerbalken für die holographisch rekonstruierte Amplitude und Phase zu erhalten. Zum Zweiten wird der Streuprozess selber untersucht, was zu einer Dichtematrixbeschreibung der holographischen Messung führt. Den Rahmen dieser Untersuchungen liefern die Gesetze der Quantenelektrodynamik. Relativistische Phänomene wie die Retardierung elektromagnetischer Felder, welche zwischen Strahlelektron und Objekt ausgetauscht werden, oder Spin-Bahn Kopplung des Strahlelektrons werden quantifiziert, wobei letzteres als unwichtig für TEM eingestuft werden konnte. Die Dekohärenz der Elektronenwelle durch die statistische Kopplung an das thermisch bewegte Kristallgitter der Keramik wird mit einem neu entwickelten Algorithmus beschrieben, welcher insbesondere die genaue Quantifizierung der elastischen Streuung an schweren Elementen erlaubt. Ein weiterer neuer Streualgorithmus, welcher elastische und inelastische Streuung kombiniert, wird entwickelt und angewendet, um spezifische Streukontraste von Dipolübergängen vorauszusagen und das altbekannte Problem der Kontrastdiskrepanz zwischen simulierten und experimentellen Bildkontrasten zu diskutieren. Um eine anwenderfreundliche und fortdauernde Anwendung der Erkenntnisse zu ermöglichen, wurde das Softwarepaket SEMI geschrieben, welches die Simulation elastischer und inelastischer Streuprozesse und des nachfolgenden Bildgebungsprozesses innerhalb verschiedener Näherungen ermöglicht und die neu entwickelten Algorithmen beinhaltet. Zum Dritten kommen dichtefunktionalbasierte Festkörperrechenmethoden zur Anwendung um charakteristische elektromagnetische Felder, wie sie beispielsweise an Domänengrenzen entstehen, innerhalb typischer ferroischer Keramiken wie BaTiO3 oder BiFeO3 zu identifizieren und zu quantifizieren und gleichzeitig Modelle zu entwickeln, welche Observablen des (holographischen) Experiments mit Charakteristika des Materials, beispielsweise den Ordnungsparamtern, korrelieren. Dies ist besonders wichtig, wenn statische elektromagnetische Felder keinen direkten Zugang zu den Ordnungsparametern, wie zum Beispiel die ferroelektrische Polarisation, liefern; beispielsweise besteht innerhalb linearer Antworttheorie die Möglichkeit, atomare Positionen mit der elektrischen Polarisation zu korrelieren. Ein Softwarepaket wurde entwickelt, welches die automatische Bestimmung der Atompositionen und der daraus resultierenden lokalen Polarisation ermöglicht. In einem vierten Schritt wurden mit Hilfe des überarbeiteten holographischen Bildgebungsprozesses in Kombination mit den aus der Streutheorie gewonnenen Erkenntnissen holographische Experimente analysiert und als Input für die mit Hilfe der Festkörpertheorie entwickelten Modelle genutzt, um quantitative Informationen über raumferroische Materialien wie BaTiO3 und PbTiO3 und kompliziertere Anordnungen wie Domänengrenzen in BiFeO3 und KnbO3 zu gewinnen. Es konnte festgestellt werden, dass spezifische atomare Verschiebungen, welche charakteristisch für Ferroelektrika sind, die zuverlässigste quantitative Information über die Polarisation bis in den Längenbereich einiger nm liefern, wogegen kleinste Wellenmodifikationen aufgrund charakteristischer Elektronenverteilungen innerhalb der Keramik mit Hilfe von Elektronenholographie nur unzureichend interpretierbar sind. Das lineare Antwortprogramm, welches die Atompositionen über Bornsche effektive Ladungen mit ferroelektrischer Polarisation korreliert, wurde erfolgreich angewendet, um Größeneffekte und Ausdehnungen von Abschirmschichten und Polarisationladungen in nichtferroelektrisch/ferroelektrischen Schichtsystemen zu bestimmen. Abschließend widmet sich ein spezieller Abschnitt der Auswertung 3D elektromagnetischer Felder mit Hilfe der elektronenholographischen Tomographie, was die Voraussetzung für die Charakterisierung von noch komplizierteren 3D Domänenwandanordnungen liefert. Da die Möglichkeiten dieser Technik durch den holographischen Rekonstruktionsfehler und spezifisch tomographische Probleme noch beschränkt sind, liegt der Schwerpunkt dieses Abschnitts in der Charakterisierung und Verbesserung des tomographischen Rekonstruktionsprozesses. Es wird eine singulärwertbasierte Rekonstruktionsmethode entwickelt, welche die Quantifizierung und Kontrolle des Rekonstruktionsfehlers ermöglicht. Außerdem wird die Vektorfeldrekonstruktion erweitert, um magnetische Vektorfelder, welche über das Rekonstruktionsvolumen hinausragen, zu behandeln.

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.

This thesis explores the theory of electron energy-loss spectroscopy (EELS) in atomic resolution electron microscopy. / The first unequivocal evidence of the effective nonlocal potential in momentum-transfer-resolved EELS is presented. For suitable geometries, the nonlocal potential can be well approximated by a local potential. In scanning transmission electron microscopy (STEM) the validity of this is mainly influenced by the detector size and, contrary to conventional wisdom, a thin annular detector does not allow direct image interpretation. It is found that the best way to ensure the potential is well approximated by a local potential is to use a detector with a large collection angle. / To simplify computation and interpretation it is desirable to make the single-channelling approximation. In this approximation only the elastic scattering of the probe before the ionisation event is modelled. It is shown how this approximation breaks down for the small detectors used in momentum-transfer-resolved EELS and this is confirmed with experimental results. Double-channelling calculations, where the channelling of the probe both before and after the ionisation event are modelled, can also be simulated. An alternative approximation for small detectors that includes double channelling and is more applicable for momentum-transfer-resolved EELS is also presented. / Beyond chemical information, the fine structure of an absorption edge gives bonding and electronic information. Incorporating fine structure into channelling theory allows the exploration of the effects of channelling on fine structure. The weighting of the two different spectra in graphite, as a function of incident probe tilt in momentum-transfer-resolved EELS, is calculated using double-channelling simulations. This is combined with experimental data and multivariate statistical analysis to extract the two physical spectra, greatly simplifying the analysis of a large data set. / The effect of the nonlocal potential and channelling on site-specific electronic structure analysis by channelling EELS is examined. It is found that using a large on-axis detector can make the interaction effectively local, leading to a greater change in the spectra as a function of sample tilt. Alternatively offsetting the detector can achieve similar results but at the cost of greater statistical noise. Channelling calculations were combined with the program FEFF and the full energy differential cross section was calculated from first principles for the aluminium K edge as a function of sample tilt in nickel aluminate spinel. Qualitative agreement with experiment was found but quantitative agreement will require further investigation. / The theory of fine structure in STEM was examined, using strontium titanate to see how the high spatial resolution of STEM can be used in conjunction with energy-loss near-edge spectroscopy measurements. The possibility of imaging unoccupied electron molecular orbitals using STEM was also examined.

214 pages / The Fock-Tani (unitary) transformation of the second- quantized Hamiltonian gives a representation which treats reactants and products symmetrically, and composites exactly. Each term in the Fock-Tani potential corresponds to a specific physical process and contains terms orthogonalizing continuum states to the bound states. The difficulty in carrying out this transformation can be lessened by working in a center of mass system, giving (n-1) reduced mass particles. After a general analysis of such systems, the Fock- Tani transformations in the 3→2-body case are carried out for the reactions a⁺+(b⁺c⁻)→(a⁺c⁻)+b⁺ and a⁻+(b⁺c⁻)→(a⁻b⁺)+c⁻. It is found that for (2) the transformation in the symmetrical reduced mass system can easily be carried out, but the Jacobi reduced mass system requires the more complicated d-matrix approach. This transformation has not yet been attempted in the full 3-body system but is likely to be as difficult as that for (1). First order differential and total cross sections are computed for resonant charge transfer in (1) for a proton- hydrogen initial state. The Fock-Tani T-matrix for the initial-state Jacobi system is found to be identical to that for the full 3-body system. That for the symmetrical reduced mass system gives an error of order l/mprot in the incident wave vector. A comparison of the Jacobi version and a previous special case Fock-Tani transformation, where the proton mass is taken as infinite, is also made. Cross sections for (ls→ls) positronium formation in positron-hydrogen collisions, calculated using the same program as for the proton-hydrogen case, are found to disagree with the previous Fock-Tani result, probably due to lack of convergence of the previous result. Cross sections for reactions (1) involving muons in hydrogenic isotopes (of interest in quantum electrodynamics and catalyzed fusion) are also calculated. Finally, extension of the results to higher order is considered. Polarized Schrodinger wave functions for a system containing a hydrogenic atom and a fully kinetic external charge are found to first order. These would be used in the Fock-Tani matrix elements to account for some initial- and final-state effects. Calculations of distorted second-quantized states and second and third order T-matrix elements are also outlined.

Abstract This dissertation is a combination of four articles on the topic of scattering problems for a biharmonic operator. The operator of interest has two coefficients which may be complex-valued and singular. Each of the articles concerns a different aspect of the problem. Namely, the first article discusses the direct scattering problem in higher dimensions and culminates in a proof of Saito's formula, which yields a uniqueness result for the inverse scattering problem. The second paper is about a backscattering problem in two and three dimensions. We prove that the inverse Born approximation can be used to recover the singularities in the coefficients of the operator. The third article fills in an answer to the question about recovering the complex-valued coefficients in three dimensions that was left open in the second article. The final article studies the inverse scattering problem on the line for a quasi-linear operator. / Tiivistelmä Väitöskirjatyö koostuu neljästä artikkelista, jotka käsittelevät sirontaongelmia biharmoniselle operaattorille. Työn kohteena olevalla operaattorilla on kaksi kerrointa, jotka voivat olla kompleksiarvoisia ja singulaarisia. Kukin artikkeli käsittelee sirontaongelmaa eri näkökulmasta. Ensimmäinen artikkeli koostuu pääasiassa suorasta sirontateoriasta korkeammissa ulottuvuuksissa huipentuen lopulta Saiton kaavan todistukseen, jonka seurauksena saadaan yksikäsitteisyystulos käänteiselle sirontaongelmalle. Toisen artikkelin aiheena on takaisinsirontaongelma kahdessa ja kolmessa ulottuvuudessa. Todistamme, että käänteistä Bornin approksimaatiota voidaan käyttää paikantamaan kertoimien mahdolliset singulariteetit. Kolmas artikkeli vastaa toisessa artikkelissa avoimeksi jääneeseen kysymykseen kompleksiarvoisien kertoimien rekonstruoimisesta kolmessa ulottuvuudessa. Viimeisessä artikkelissa tutkitaan käänteistä sirontaongelmaa kvasilineaariselle operaattorille yhdessä ulottuvuudessa.

The thesis deals with quantum reaction dynamics of three-particle systems. The thesis summarizes main theoretical results about three-body problem in quantum mechanics. A simple two dimensional model of three-particle system corresponding to atom-diatom collision was studied as a part of this thesis. The model allows for vibrational excitation and reaction processes. Solution based on distorted-wave method and Schwinger variational principle is proposed to solve the model. Proposed method of solution is then applied to a system of three coupled Morse potential energy surfaces. Probabilities of possible proces- ses depending on energy of incoming particle were calculated using the proposed method of solution for two variants of coupled Morse PESs. 1