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The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control. / Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716 / TESIS / Premios Extraordinarios de tesis doctorales

The discipline of component-based modeling and simulation offers promising gains including reduction in development cost, time, and system complexity. This paradigm is very profitable as it promotes the use and reuse of modular components and is auspicious for effective development of complex simulations. It however is confronted by a series of research challenges when it comes to actually practice this methodology. One of such important issue is Composability verification. In modeling and simulation (M&amp;S), composability is the capability to select and assemble components in various combinations to satisfy specific user requirements. Therefore to ensure the correctness of a composed model, it is verified with respect to its requirements specifications.There are different approaches and existing component modeling frameworks that support composability however in our observation most of the component modeling frameworks possess none or weak built-in support for the composability verification. One such framework is Base Object Model (BOM) which fundamentally poses a satisfactory potential for effective model composability and reuse. However it falls short of required semantics, necessary modeling characteristics and built-in evaluation techniques, which are essential for modeling complex system behavior and reasoning about the validity of the composability at different levels.In this thesis a comprehensive verification framework is proposed to contend with some important issues in composability verification and a verification process is suggested to verify composability of different kinds of systems models, such as reactive, real-time and probabilistic systems. With an assumption that all these systems are concurrent in nature in which different composed components interact with each other simultaneously, the requirements for the extensive techniques for the structural and behavioral analysis becomes increasingly challenging. The proposed verification framework provides methods, techniques and tool support for verifying composability at its different levels. These levels are defined as foundations of a consistent model composability. Each level is discussed in detail and an approach is presented to verify composability at that level. In particular we focus on theDynamic-Semantic Composability level due to its significance in the overallcomposability correctness and also due to the level of difficulty it poses in theprocess. In order to verify composability at this level we investigate the application ofthree different approaches namely (i) Petri Nets based Algebraic Analysis (ii) ColoredPetri Nets (CPN) based State-space Analysis and (iii) Communicating SequentialProcesses based Model Checking. All the three approaches attack the problem ofverifying dynamic-semantic composability in different ways however they all sharethe same aim i.e., to confirm the correctness of a composed model with respect to itsrequirement specifications. Beside the operative integration of these approaches inour framework, we also contributed in the improvement of each approach foreffective applicability in the composability verification. Such as applying algorithmsfor automating Petri Net algebraic computations, introducing a state-space reductiontechnique in CPN based state-space analysis, and introducing function libraries toperform verification tasks and help the molder with ease of use during thecomposability verification. We also provide detailed examples of using each approachwith different models to explain the verification process and their functionality.Lastly we provide a comparison of these approaches and suggest guidelines forchoosing the right one based on the nature of the model and the availableinformation. With a right choice of an approach and following the guidelines of ourcomponent-based M&amp;S life-cycle a modeler can easily construct and verify BOMbased composed models with respect to its requirement specifications. / <p>Overseas Scholarship for PHD in selected Studies Phase II Batch I</p><p>Higher Education Commision of Pakistan.</p><p>QC 20130224</p>

The subject matter of this dissertation is the design, analysis and practical implementation of a new numerical method to approximate the eigenvalues and eigenfunctions of regular Sturm–Liouville problems, given in Liouville’s normal form, defined on compact intervals, with self-adjoint separated boundary conditions. These are classical problems in computational mathematics which lie on the interface between numerical analysis and spectral theory, with important applications in physics and chemistry, not least in the approximation of energy levels and wave functions of quantum systems. Because of their great importance, many numerical algorithms have been proposed over the years which span a vast and diverse repertoire of techniques. When compared with previous approaches, the principal advantage of the numerical method proposed in this dissertation is that it is accompanied by error bounds which: (i) hold uniformly over the entire eigenvalue range, and, (ii) can attain arbitrary high-order. This dissertation is composed of two parts, aggregated according to the regularity of the potential function. First, in the main part of this thesis, this work considers the truncation, discretization, practical implementation and MATLAB software, of the new approach for the classical setting with continuous and piecewise analytic potentials (Ramos and Iserles, 2015; Ramos, 2015a,b,c). Later, towards the end, this work touches upon an extension of the new ideas that enabled the truncation of the new approach, but instead for the general setting with absolutely integrable potentials (Ramos, 2014).