<p>Abstract: Considering the Schrödinger equation $\Delta_x u = i\partial{u}/\partial{t}$, we have a solution $u$ on the form $$u(x, t)= (2\pi)^{-n} \int_{\RR} {e^{i x\cdot \xi}e^{it|\xi|^2}\widehat{f}(\xi)}\, d \xi, x \in \RR, t \in \mathbf{R}$$ where $f$ belongs to the Sobolev space. It was shown by Sjögren and Sjölin, that assuming $\gamma : \mathbf{R}_+ \rightarrow \mathbf{R}_+ $ being a strictly increasing function, with $\gamma(0) = 0$ and $u$ and $f$ as above, there exists an $f \in H^{n/2} (\RR)$ such that $u$ is continuous in $\{ (x, t); t>0 \}$ and $$\limsup_{(y,t)\rightarrow (x,0),|y-x|<\gamma (t), t>0} |u(y,t)|= + \infty$$ for all $x \in \RR$. This theorem was proved by choosing $$\widehat{f}(\xi )=\widehat{f_a}(\xi )= | \xi | ^{-n} (\log | \xi |)^{-3/4} \sum_{j=1}^{\infty} \chi _j(\xi)e^{- i( x_{n_j} \cdot \xi + t_j | \xi | ^a)}, \, a=2,$$ where $\chi_j$ is the characteristic function of shells $S_j$ with the inner radius rapidly increasing with respect to $j$. The purpose of this essay is to explain the proof given by Sjögren and Sjölin, by first showing that the theorem is true for $\gamma (t)=t$, and to investigate the result when we use $$S^a f_a (x, t)= (2 \pi)^{-n}\int_{\RR} {e^{i x\cdot \xi}e^{it |\xi|^a}\widehat{f_a}(\xi)}\, d \xi$$ instead of $u$.</p>

Abstract: Considering the Schrödinger equation $\Delta_x u = i\partial{u}/\partial{t}$, we have a solution $u$ on the form $$u(x, t)= (2\pi)^{-n} \int_{\RR} {e^{i x\cdot \xi}e^{it|\xi|^2}\widehat{f}(\xi)}\, d \xi, x \in \RR, t \in \mathbf{R}$$ where $f$ belongs to the Sobolev space. It was shown by Sjögren and Sjölin, that assuming $\gamma : \mathbf{R}_+ \rightarrow \mathbf{R}_+ $ being a strictly increasing function, with $\gamma(0) = 0$ and $u$ and $f$ as above, there exists an $f \in H^{n/2} (\RR)$ such that $u$ is continuous in $\{ (x, t); t&gt;0 \}$ and $$\limsup_{(y,t)\rightarrow (x,0),|y-x|&lt;\gamma (t), t&gt;0} |u(y,t)|= + \infty$$ for all $x \in \RR$. This theorem was proved by choosing $$\widehat{f}(\xi )=\widehat{f_a}(\xi )= | \xi | ^{-n} (\log | \xi |)^{-3/4} \sum_{j=1}^{\infty} \chi _j(\xi)e^{- i( x_{n_j} \cdot \xi + t_j | \xi | ^a)}, \, a=2,$$ where $\chi_j$ is the characteristic function of shells $S_j$ with the inner radius rapidly increasing with respect to $j$. The purpose of this essay is to explain the proof given by Sjögren and Sjölin, by first showing that the theorem is true for $\gamma (t)=t$, and to investigate the result when we use $$S^a f_a (x, t)= (2 \pi)^{-n}\int_{\RR} {e^{i x\cdot \xi}e^{it |\xi|^a}\widehat{f_a}(\xi)}\, d \xi$$ instead of $u$.

Halo nuclei are among the strangest nuclear structures. They are viewed as a core containing most of the nucleons surrounded by one or two loosely bound nucleons. These have a high probability of presence at a large distance from the core. Therefore, they constitute a sort of halo surrounding the other nucleons. The core, remaining almost unperturbed by the presence of the halo is seen as a usual nucleus. <P> The Coulomb breakup reaction is one of the most useful tools to study these nuclei. It corresponds to the dissociation of the halo from the core during a collision with a heavy (high <I>Z</I>) target. In order to correctly extract information about the structure of these nuclei from experimental cross sections, an accurate theoretical description of this mechanism is necessary. <P> In this work, we present a theoretical method for studying the Coulomb breakup of one-nucleon halo nuclei. This method is based on a semiclassical approximation in which the projectile is assumed to follow a classical trajectory. In this approximation, the projectile is seen as evolving in a time-varying potential simulating its interaction with the target. This leads to the resolution of a time-dependent Schrödinger equation for the projectile wave function. <P> In our method, the halo nucleus is described with a two-body structure: a pointlike nucleon linked to a pointlike core. In the present state of our model, the interaction between the two clusters is modelled by a local potential. <P> The main idea of our method is to expand the projectile wave function on a three-dimensional spherical mesh. With this mesh, the representation of the time-dependent potential is fully diagonal. Furthermore, it leads to a simple representation of the Hamiltonian modelling the halo nucleus. This expansion is used to derive an accurate evolution algorithm. <P> With this method, we study the Coulomb breakup of three nuclei: ¹¹Be, ¹⁵C and ⁸B. ¹¹Be is the best known one-neutron halo nucleus. Its Coulomb breakup has been extensively studied both experimentally and theoretically. Nevertheless, some uncertainty remains about its structure. The good agreement between our calculations and recent experimental data suggests that it can be seen as a <I>s1/2</I> neutron loosely bound to a ¹⁰Be core in its 0⁺ ground state. However, the extraction of the corresponding spectroscopic factor have to wait for the publication of these data. <P> ¹⁵C is a candidate one-neutron halo nucleus whose Coulomb breakup has just been studied experimentally. The results of our model are in good agreement with the preliminary experimental data. It seems therefore that ¹⁵C can be seen as a ¹⁴C core in its 0⁺ ground state surrounded by a <I>s1/2</I> neutron. Our analysis suggests that the spectroscopic factor corresponding to this configuration should be slightly lower than unity. <P> We have also used our method to study the Coulomb breakup of the candidate one-proton halo nucleus ⁸B. Unfortunately, no quantitative agreement could be obtained between our results and the experimental data. This is mainly due to an inaccuracy in the treatment of the results of our calculations. Accordingly, no conclusion can be drawn about the pertinence of the two-body model of ⁸B before an accurate reanalysis of these results. <P> In the future, we plan to improve our method in two ways. The first concerns the modelling of the halo nuclei. It would be indeed of particular interest to test other models of halo nuclei than the simple two-body structure used up to now. The second is the extension of this semiclassical model to two-neutron halo nuclei. However, this cannot be achieved without improving significantly the time-evolution algorithm so as to reach affordable computational times.

Accurate numerical solution of time-dependent, high-dimensional partial differential equations (PDEs) usually requires efficient numerical techniques and massive-scale parallel computing. In this thesis, we implement and evaluate discretization schemes suited for PDEs of higher dimensionality, focusing on high order of accuracy and low computational cost. Spatial discretization is particularly challenging in higher dimensions. The memory requirements for uniform grids quickly grow out of reach even on large-scale parallel computers. We utilize high-order discretization schemes and implement adaptive mesh refinement on structured hyperrectangular domains in order to reduce the required number of grid points and computational work. We allow for anisotropic (non-uniform) refinement by recursive bisection and show how to construct, manage and load balance such grids efficiently. In our numerical examples, we use finite difference schemes to discretize the PDEs. In the adaptive case we show how a stable discretization can be constructed using SBP-SAT operators. However, our adaptive mesh framework is general and other methods of discretization are viable. For integration in time, we implement exponential integrators based on the Lanczos/Arnoldi iterative schemes for eigenvalue approximations. Using adaptive time stepping and a truncated Magnus expansion, we attain high levels of accuracy in the solution at low computational cost. We further investigate alternative implementations of the Lanczos algorithm with reduced communication costs. As an example application problem, we have considered the time-dependent Schrödinger equation (TDSE). We present solvers and results for the solution of the TDSE on equidistant as well as adaptively refined Cartesian grids. / eSSENCE

This thesis explores the dynamics of a heteronuclear diatomic molecular ion, possessing a permanent electric dipole moment, µ, which is trapped in a linear Paul trap and can interact with an off-resonance laser field. To build our model we use the rigid-rotor approximation, where the dynamics of the molecular ion are limited to its translational and rotational motions of the center-of-mass. These dynamics are investigated by carrying out suitable numerical calculations. To introduce our numerical methods, we divide our research topic into two different subjects. First, we ignore the rotational dynamics of the ion by assuming µ = 0. By this assumption, the system resembles an atomic ion, which mainly exhibits translational motion for its center of the mass when exposed to an external trapping field. To study this translational behavior, we implement full-quantum numerical simulations, in which a wave function is attributed to the ion. Finally, we study the quantum dynamics of the mentioned wave packet and we compare our results with those obtained classically. In the latter case, we keep the permanent dipole moment of the ion and we study the probable effects of the interaction between the dipole moment and the trapping electric field, on both the translational and the rotational dynamics of the trapped molecular ion. In order to study these dynamics, we implement both classical and semi-classical numerical simulations. In the classical method, the rotational and the translational motions of the center of mass of the ion are obtained via classical equations of motion. On the other hand, in the semi-classical method, while the translational motion of the center-of-mass is still obtained classically, the rotation is treated full-quantum mechanically by considering the rotational wave function of the ion. In the semi-classical approach, we mainly study the probable couplings between the rotational states of the molecular ion, due to the interaction of the permanent dipole moment with the trapping electric field. In the end, we also present a semi-classical model, where the trapped molecular ion interacts with an off-resonance laser field.

We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrödinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable. The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space. / eSSENCE

Eine Methode zur Lösung der zeitabhängigen Schrödingergleichung (engl. time-dependent Schrödinger equation, TDSE) wurde entwickelt, welche das Verhalten der Elektronenbewegung in Molekülen beschreibt, die ultrakurzen, intensiven Laserpulsen ausgesetzt werden. Die zeitabhängigen elektronischen Wellenfunktionen werden durch eine Superposition von feldfreien Eigenzuständen beschrieben, welche auf zwei Weisen berechnet werden. Im ersten Ansatz , welcher auf Zweielektronen-Systeme wie H$_2$ anwendbar ist, werden die voll korrelierten feldfreien Eigenzustände in voller Dimensionalität in einem Konfigurations-Wechselwirkungs Verfahren (engl. configuration interaction, CI) bestimmt, wobei die Einelektron-Basisfunktionen mit B-Splines beschrieben werden. Im zweiten Verfahren, welches sogar auf größere Moleküle anwendbar ist, werden die feldfreien Eigenzustände in der Näherung eines aktiven Elektrons (engl. single active electron, SAE) mit Verwendung der Dichtefunktionaltheorie (DFT) bestimmt. Im Allgemeinen kann die Methode zum Auffinden der zeitabhängigen Lösung in zwei Schritte, dem Auffinden der feldfreien Eigenzustände und einer Zeitpropagation in Abhängigkeit der Laserpuls-Parameter, unterteilt werden. Die Gültigkeit der SAE Näherung ist überprüft und die Ergebnisse für grund und erste angeregte zustand der Wasserstoff-Molekül werden vorgestellt. Die Ergebnisse für einige größere Moleküle innerhalb der SAE Angleichung werden ebenfalls gezeigt. / A method for solving the time-dependent Schrödinger equation (TDSE) describing the electronic motion of the molecules exposed to very short intense laser pulses has been developed. The time-dependent electronic wavefunction is expanded in terms of a superposition of field-free eigenstates. The field-free eigenstates are calculated in two ways. In the first approach, which is applicable to two electron systems like hydrogen molecule, fully correlated field-free eigenstates are obtained in complete dimensionality using configuration-interaction calculation where the one-electron basis functions are built from B-splines. In the second approach, which is even applicable to larger molecules, the field-free eigenstates are calculated within the single-active-electron (SAE) approximation using density functional theory. In general, the method can be divided into two parts, in the first part the field-free eigenstates are calculated and then in the second part a time propagation for the laser pulse parameters is performed. Using these methods the validity of SAE approximation is tested and the results for the ground and first excited state of hydrogen molecule are presented. The results for some larger molecules within the SAE approximation are also shown.

Ein neuer numerischer ab initio Ansatz wurde entwickelt und zur Lösung der zeitabhängigen Schrödingergleichung für zweiatomig Moleküle mit zwei Elektronen (z.B. molekularer Wasserstoff), welche einem intensiven kurzen Laserpuls ausgesetzt sind, angewandt. Die Methode basiert auf der Näherung fester Kernabstände und der nicht-relativistischen Dipolnäherung und beabsichtigt die genaue Beschreibung der beiden korrelierten Elektronen in voller Dimensionalität. Die Methode ist anwendbar für eine große Bandbreite von Laserpulsparamtern und ist in der Lage, Einfachionisationsprozesse sowohl mit wenigen als auch mit vielen Photonen zu beschreiben, sogar im nicht-störungstheoretischen Bereich. Ein entscheidender Vorteil der Methode ist ihre Fähigkeit, die Reaktion von Molekülen mit beliebiger Orientierung der molekularen Achse im Bezug auf das linear polarisierte Laserfeld in starken Feldern zu beschreiben. Dementsprechend berichtet diese Arbeit von der ersten erfolgreichen orientierungsabhängigen Analyse der Multiphotonenionisation von H2, welche mit Hilfe einer numerischen Behandlung in voller Dimensionalität durchgeführt wurde. Neben der Erforschung des Bereichs weniger Photonen wurde eine ausführliche numerische Untersuchung der Ionisation durch ultrakurze frequenzverdoppelte Titan:Saphir-Laserpulse (400 nm) präsentiert. Mit Hilfe einer Serie von Rechnungen für verschiedene Kernabstände wurden die totalen Ionisationsausbeuten für H2 und D2 in ihren Vibrationsgrundzuständen sowohl für parallele als auch für senkrechte Ausrichtung erhalten. Eine weitere Serie von Rechnungen für 800nm Laserpulse wurde benutzt, um ein weitverbreitetes einfaches Interferenzmodel zu falsifizieren. Neben der Diskussion der numerischen ab initio Methode werden in dieser Arbeit verschiedene Aspekte im Bezug auf die Anwendung der Starkfeldnäherung für die Erforschung der Reaktion eines atomaren oder molekularen Systems auf ein intensives Laserfeld betrachtet. / A novel ab initio numerical approach is developed and applied that solves the time-dependent Schrödinger equation describing two-electron diatomic molecules (e.g. molecular hydrogen) exposed to an intense ultrashort laser pulse. The method is based on the fixed-nuclei and the non-relativistic dipole approximations and aims to accurately describe both correlated electrons in full dimensionality. The method is applicable for a wide range of the laser pulse parameters and is able to describe both few-photon and many-photon single ionization processes, also in a non-perturbative regime. A key advantage of the method is its ability to treat the strong-field response of the molecules with arbitrary orientation of the molecular axis with respect to the linear-polarized laser field. Thus, this work reports on the first successful orientation-dependent analysis of the multiphoton ionization of H2 performed by means of a full-dimensional numerical treatment. Besides the investigation of few-photon regime, an extensive numerical study of the ionization by ultrashort frequency-doubled Ti:sapphire laser pulses (400 nm) is presented. Performing a series of calculations for different internuclear separations, the total ionization yields of H2 and D2 in their ground vibrational states are obtained for both parallel and perpendicular orientations. A series of calculations for 800nm laser pulses are used to test a popular simple interference model. Besides the discussion of the ab initio numerical method, this work considers different aspects related to the application of the strong-field approximation (SFA) for investigation of a strong-field response of an atomic and molecular system. Thus, a deep analysis of the gauge problem of SFA is performed and the quasistatic limit of the velocity-gauge SFA ionization rates is derived. The applications of the length gauge SFA are examined and a recently proposed generalized Keldysh theory is criticized.

Das Ionisierungsverhalten kleiner Moleküle (insbesondere H2 und NH3) in intensiven, ultrakurzen Laserfeldern wird theoretisch untersucht. Das Hauptaugenmerk liegt dabei auf dem Einfluss der Kerndynamik. Zunächst wird das Ionisierungsverhalten des H2-Moleküls bei eingefrorener Kernschwingung untersucht. Bereits im Rahmen dieser Näherung kann im Mehrphotonenregime ein zuvor beobachteter Zusammenbruch der Näherung im Gleichgewichtsabstand festgehaltener Kerne erklärt werden. Weiterhin wird der Übergang vom Mehrphotonen zum quasistatischen Ionisierungsregime für 800-nm-Laserfelder untersucht. Eine neuartige Methode zur Beschreibung der korrelierten Schwingungs- und Elektronendynamik des H2-Moleküls (7D) wird entwickelt. Mit dieser Methode wird schließlich der Einfluss der Kernbewegung während des Laserfeldes auf das Ionisierungsverhalten untersucht. Es wird ein sichtbarer Einfluss auf den zuvor diskutierten Zusammenbruch der Näherung festgehaltener Kerne beobachtet. Dies gilt ebenfalls für einen vor kurzem experimentell beobachteten Isotopeneffekt in der Ionisierung der Moleküle H2 vs. D2 untersucht. Im zweiten Teil der Arbeit wird das Ionisierungsverhalten des NH3-Moleküls untersucht. Die Möglichkeit, die Kerngeometrieabhängigkeit zur Erzeugung und Messung von Schwingungswellenpaketen im neutralen NH3-Molekül mittels Lochfraß auszunutzen, wird untersucht. Das erwartete Schwingungsverhalten und die dafür optimalen Laserparameter werden aufgezeigt. Zusätzlich wird die Möglichkeit des Filmens eines tunnelnden Kernwellenpakets im Doppelmuldenpotential entlang der Schwingungskoordinate untersucht. In der Tat sollte die Verwendung extrem kurzer Laserfelder das Drehen eines Echtzeit-Filmes dieses quantenmechanischen Tunnelprozesses ermöglichen. Abschließend werden die Winkelabhängigkeit der Ionisierungswahrscheinlichkeit von NH3 (ähnelt Orbitalgeometrie) sowie elliptisch polarisierte Laserfelder untersucht. / The ionization behavior of small molecules (especially H2 and NH3) exposed to intense, ultrashort laser fields is investigated theoretically. The focus lies on the influence of nuclear dynamics on this ionization behavior. The ionization behavior of the H2 molecule is first examined within the frozen-nuclei approximation. A previously reported pronounced breakdown of the fixed-nuclei approximation can be explained already within this level of approximation. Furthermore, the transition from the multiphoton to the quasistatic ionization regime is studied for 800 nm laser pulses. A novel approach for the correlated description of the electronic-vibrational motion of the H2 molecule (7D) is developed. The influence of vibrational dynamics during the laser field on the ionization behavior is investigated using this method. A pronounced difference on the previously discussed breakdown of the fixed-nuclei approximation is observed. The vibrational dynamics also lead to a notable change for a recently experimentally observed isotope effect in the ionization of the molecular isotopes H2 vs. D2. The ionization behavior of the NH3 molecule is studied in the second part of this thesis. The possibility to exploit the geometry dependence of the ionization yield in order to create and measure vibrational wave packets in the neutral NH3 molecule via Lochfraß is explored. The expected vibrational dynamics and the optimal laser parameters to observe this effect are demonstrated. Furthermore, the possibility to shoot a "movie" of a tunneling wave packet in the double-well potential along the vibrational coordinate is investigated. Indeed, extremely short laser fields should allow creating a real-time movie of the quantum-mechanical tunneling process. Finally, the orientation dependence of the ionization yield of the NH3 molecule (reflecting the orbital shape) and elliptically polarized laser fields are studied.

The time-dependent Schrödinger equation (TDSE) models the quantum nature of molecular processes.  Numerical simulations based on the TDSE help in understanding and predicting the outcome of chemical reactions. This thesis is dedicated to the derivation and analysis of efficient and reliable simulation tools for the TDSE, with a particular focus on models for the interaction of molecules with time-dependent electromagnetic fields. Various time propagators are compared for this setting and an efficient fourth-order commutator-free Magnus-Lanczos propagator is derived. For the Lanczos method, several communication-reducing variants are studied for an implementation on clusters of multi-core processors. Global error estimation for the Magnus propagator is devised using a posteriori error estimation theory. In doing so, the self-adjointness of the linear Schrödinger equation is exploited to avoid solving an adjoint equation. Efficiency and effectiveness of the estimate are demonstrated for both bounded and unbounded states. The temporal approximation is combined with adaptive spectral elements in space. Lagrange elements based on Gauss-Lobatto nodes are employed to avoid nondiagonal mass matrices and ill-conditioning at high order. A matrix-free implementation for the evaluation of the spectral element operators is presented. The framework uses hybrid parallelism and enables significant computational speed-up as well as the solution of larger problems compared to traditional implementations relying on sparse matrices. As an alternative to grid-based methods, radial basis functions in a Galerkin setting are proposed and analyzed. It is found that considerably higher accuracy can be obtained with the same number of basis functions compared to the Fourier method. Another direction of research presented in this thesis is a new algorithm for quantum optimal control: The field is optimized in the frequency domain where the dimensionality of the optimization problem can drastically be reduced. In this way, it becomes feasible to use a quasi-Newton method to solve the problem. / eSSENCE