Indiana University-Purdue University Indianapolis (IUPUI) / Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semi-classical models with mode selective losses, and lossy quantum systems. The rapidly growing research on these systems has mainly focused on the wide range of novel functionalities they demonstrate. In this thesis, I intend to present some intriguing properties of a class of open systems which possess parity (P) and time-reversal (T) symmetry with a theoretical background, accompanied by the experimental platform these are realized on. These systems show distinct regions of broken and unbroken symmetries separated by a special phase boundary in the parameter space. This separating boundary is called the PT-breaking threshold or the PT transition threshold. We investigate non-Hermitian systems in two settings: tight binding lattice models, and electrical circuits, with the help of theoretical and numerical techniques. With lattice models, we explore the PT-symmetry breaking threshold in discrete realizations of systems with balanced gain and loss which is determined by the effective coupling between the gain and loss sites. In one-dimensional chains, this threshold is maximum when the two sites are closest to each other or the farthest. We investigate the fate of this threshold in the presence of parallel, strongly coupled, Hermitian (neutral) chains, and find that it is increased by a factor proportional to the number of neutral chains. These results provide a surprising way to engineer the PT threshold in experimentally accessible samples. In another example, we investigate the PT-threshold for a one-dimensional, finite Kitaev chain—a prototype for a p-wave superconductor— in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, onsite potential, and the distance between the gain and loss sites. In addition to a robust, non-local threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band-structure of the Hermitian Kitaev model. Finally, with electrical circuits, we propose a protocol to study the properties of a PT-symmetric system in a single LC oscillator circuit which is contrary to the notion that these systems require a pair of spatially separated balanced gain and loss elements. With a dynamically tunable LC oscillator with synthetically constructed circuit elements, we demonstrate static and Floquet PT breaking transitions by tracking the energy of the circuit. Distinct from traditional mechanisms to implement gain and loss, our protocol enables parity-time symmetry in a minimal classical system.

Mesh free modeling techniques are a promising alternative to traditional meshed methods for solving computational fluid dynamics problems. These techniques aim to solve for the field variable using solely the values of nodes and therefore do not require the generation of a mesh. This results in a process that can be much more reliably automated and is therefore attractive. Radial basis functions (RBFs) are one type of "meshless" method that has shown considerable growth in the past 50 years. Using these RBFs to directly solve a partial differential equation is known as Kansa's method and has been used to successfully solve many flow problems. The problem with Kansa's method is that there is no formal guarantee that its solution matrix will be non-singular. More recently, an expansion on Kansa's method was proposed that incorporates the boundary and PDE operators into the solution of the field variable. This method, known as Hermitian method, has been shown to be non-singular provided certain nodal criteria are met. This work aims to perform a comparison between Kansa and Hermitian methods to aid in future selection of a method. These two methods were used to solve steady and transient one-dimensional convection-diffusion problems. The methods are compared in terms of accuracy (error) and computational complexity (conditioning number) in order to evaluate overall performance. Results suggest that the Hermitian method does slightly outperform Kansa method at the cost of a more ill-conditioned collocation matrix.

Matter on a length scale comparable to that of a chemical bond is governed by the theory of quantum mechanics, but quantum mechanics is a many body theory, hence for the sake of chemistry or solid state physics, finding solutions to the governing equation, Schrodinger's equation, is hopeless for all but the smallest of systems. As the number of electrons increases, the complexity of solving the equations grows rapidly without bound. One way to make progress is to treat the electrons in a system as independent particles and to attempt to capture the many-body effects in a functional of the electrons' density distribution. When this approximation is made, the resulting equation is called the Kohn-Sham equation, and instead of requiring solving for one function of many variables, it requires solving for many functions of the three spatial variables. This problem turns out to be easier than the many body problem, but it still scales cubically in the number of electrons. In this work we will explore ways of obtaining the solutions to the Kohn-Sham equation in the framework of real-space pseudopotential density functional theory. The Kohn-Sham equation itself is an eigenvalue problem, just as Schrodinger's equation. For each electron in the system, there is a corresponding eigenvector. So the task of solving the equation is to compute many eigenpairs of a large Hermitian matrix. In order to mitigate the problem of cubic scaling, we develop an algorithm to slice the spectrum into disjoint segments. This allows a smaller eigenproblem to be solved in each segment where a post-processing step combines the results from each segment and prevents double counting of the eigenpairs. The efficacy of this method depends on the use of high order polynomial filters that enhance only a segment of the spectrum. The order of the filter is the number of matrix-vector multiplication operations that must be done with the Hamiltonian. Therefore the performance of these operations is critical. We develop a scalable algorithm for computing these multiplications and introduce a new density functional theory code implementing the algorithm. / text

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Cette thèse traite de l'étude de méthodes de diagonalisation conjointe de matrices complexes, en vue de la séparation de sources, que ce soit dans le domaine des télécommunications numériques ou de la radioastronomie. Après avoir présenté les motivations qui ont poussé cette étude, nous faisons un bref état de l'art dans le domaine. Le problème de la diagonalisation conjointe, ainsi que celui de la séparation de source sont rappelés, et un lien entre ces deux sujets est établi. Par la suite, plusieurs algorithmes itératifs sont développés. Dans un premier temps, des méthodes utilisant une mise à jour de la matrice de séparation, de type gradient, sont présentées. Elles sont basées sur des approximations judicieuses du critère considéré. Afin d'améliorer la vitesse de convergence, une méthode utilisant un calcul du pas optimal est présentée, et plusieurs variantes de ce calcul, basées sur les approximations faites précédemment, sont développées. Deux autres approches sont ensuite introduites. La première détermine la matrice de séparation de manière analytique, en calculant algébriquement les termes composant la matrice de mise à jour par paire à partir d'un système d'équations linéaire. La deuxième estime récursivement la matrice de mélange, en se basant sur une méthode de moindres carrés alternés. Afin d'améliorer la vitesse de convergence, une recherche de pas d'adaptation linéaire est proposée. Ces méthodes sont alors validées sur un problème de diagonalisation conjointe classique. Puis les algorithmes sont appliqués à la séparation de sources de signaux de télécommunication numérique, en utilisant des statistiques d'ordre deux ou supérieur. Des comparaisons sont également effectuées avec des méthodes standards. La deuxième application concerne l'élimination des interférences terrestres à partir de l'estimation de l'espace associé, afin d'observer au mieux des sources cosmiques, issues de données de station LOFAR. / This thesis deals with the study of joint diagonalization of complex matrices methods for source separation, wether in the field of numerical telecommunications and radioastronomy. After having introduced the motivations that drove this study, we present a brief state-of-the-art in the field. The joint diagonalization and source separation problems are reminded, and a link between these two themes is established. Thereafter, several iterative algorithms are developed. First, methods using a gradient-like update of the separation matrix are introduced. They are based on wise approximations of the considered criterion. In order to improve the convergence speed, a method using a computation of an optimal step size is presented, and variations around this computation, based on the previously introduced approximations are done. Two other approaches are then introduced. The first one analytically determines the separation matrix, by algebraically computing the terms composing the update matrix pairwise from a linear equation system. The second one recursively estimates the mixing matrix, based on an alternating least squares method. In order to enhance the convergence speed, a seek of an enhanced line search algorithm is proposed. These methods are then validated on a classical joint diagonalization problem. Aterwards, these algorithms are applied to the source separation of numerical communication signals, while using second or higher order statistics. Comparisons are also made with well-known methods. The second application relates to elimination of rterrestrial interferences from the estimation of the associated space in order to observe at best cosmic sources from LOFAR station data.

Yip Chi Ming = 守恆及開放系統的能譜反問題 / 葉志明. / Thesis submitted in 2000. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves [244]-247). / Text in English; abstracts in English and Chinese. / Yip Chi Ming = Shou heng ji kai fang xi tong de neng pu fan wen ti / Ye Zhiming. / Abstract --- p.i / Acknowledgements --- p.ii / Contents --- p.iii / List of Figures --- p.viii / List of Tables --- p.xxi / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- The Sturm-Liouville Problem --- p.3 / Chapter 1.2 --- Historical review of inverse problems --- p.7 / Chapter 1.3 --- Conservative systems --- p.10 / Chapter 1.4 --- Open systems --- p.10 / Chapter 1.5 --- Organization of the following chapters --- p.11 / Chapter Chapter 2. --- Conservative Spectral Problem --- p.12 / Chapter 2.1 --- The system --- p.12 / Chapter 2.2 --- Properties of conservative systems --- p.13 / Chapter 2.2.1 --- Asymptotic expansion of eigenvalues --- p.14 / Chapter 2.3 --- Forward spectral problem --- p.16 / Chapter 2.3.1 --- FDM and FEM --- p.17 / Chapter 2.3.2 --- Solving transcendental equation --- p.20 / Chapter 2.4 --- Phase shift problem --- p.20 / Chapter 2.4.1 --- Square well potential --- p.22 / Chapter Chapter 3. --- Forward Spectral Problem for Open Systems --- p.25 / Chapter 3.1 --- The system --- p.26 / Chapter 3.2 --- Properties of open systems --- p.28 / Chapter 3.2.1 --- Asymptotic behaviour of QNM eigenvalues --- p.28 / Chapter 3.2.2 --- Doubling of modes --- p.33 / Chapter 3.2.3 --- Generalized norm of QNMs --- p.34 / Chapter 3.2.4 --- Completeness --- p.37 / Chapter 3.2.5 --- Eigenfunction expansion for QNMs - two component formalism --- p.39 / Chapter 3.3 --- Forward spectral problem --- p.45 / Chapter Chapter 4. --- Conservative Inverse Problem --- p.50 / Chapter 4.1 --- Sun-Young-Zou (SYZ) method --- p.51 / Chapter 4.1.1 --- Perturbative inversion --- p.53 / Chapter 4.1.2 --- The regulators (δn) --- p.54 / Chapter 4.1.3 --- Total inversion (TI) --- p.59 / Chapter 4.1.4 --- Numerical results --- p.60 / Chapter 4.2 --- Rundell and Sacks method (RS method) --- p.74 / Chapter 4.2.1 --- Completeness --- p.75 / Chapter 4.2.2 --- The integral equation --- p.78 / Chapter 4.2.3 --- Uniqueness --- p.82 / Chapter 4.2.4 --- RS formalism --- p.84 / Chapter 4.2.5 --- Numerical results and difficulties --- p.89 / Chapter 4.2.6 --- Summary --- p.110 / Chapter 4.3 --- Phase shift problem --- p.112 / Chapter 4.3.1 --- Reduction to spectral problem --- p.113 / Chapter 4.3.2 --- Modified RS algorithm for finite-range phase shift problem --- p.116 / Chapter 4.3.3 --- Discussion --- p.130 / Chapter 4.4 --- Bound states --- p.131 / Chapter Chapter 5. --- Open Inverse Problem --- p.136 / Chapter 5.1 --- SYZ method --- p.136 / Chapter 5.1.1 --- Perturbative Inversion (PI) and Total Inversion (TI) --- p.137 / Chapter 5.1.2 --- Numerical results --- p.138 / Chapter 5.1.3 --- Other choices of (δn) --- p.156 / Chapter 5.2 --- RS method --- p.158 / Chapter 5.2.1 --- The integral equation --- p.159 / Chapter 5.2.2 --- Cauchy data --- p.160 / Chapter 5.2.3 --- Completeness conjecture --- p.162 / Chapter 5.2.4 --- Numerical verification of completeness condition --- p.163 / Chapter 5.2.5 --- Inversion for Cauchy data --- p.166 / Chapter 5.2.6 --- Cauchy data on 0 < x≤ α --- p.167 / Chapter 5.2.7 --- Comparison system --- p.169 / Chapter Chapter 6. --- Conclusions and Further Studies --- p.188 / Chapter 6.1 --- Conclusions of this thesis --- p.188 / Chapter 6.2 --- Further studies --- p.189 / Chapter Appendix A. --- Singular Value Decomposition --- p.199 / Chapter Appendix B. --- Asymptotic Behaviour of Phase Shifts --- p.203 / Chapter B.1 --- Asymptotic behaviour of phase shift data --- p.203 / Chapter B.2 --- Levinson's theorem --- p.204 / Chapter Appendix C. --- Forward Problem for Conservative Systems --- p.207 / Chapter C.1 --- Finite difference method --- p.207 / Chapter C.2 --- Finite element method --- p.209 / Chapter C.2.1 --- Solving transcendental equation --- p.215 / Chapter Appendix D. --- FDM and FEM for Open Systems --- p.220 / Chapter D.1 --- Finite difference method --- p.220 / Chapter D.2 --- Finite element method --- p.222 / Chapter Appendix E. --- Asymptotic Behaviour of NM Eigenvalues --- p.226 / Chapter Appendix F. --- Asymptotic Behaviour of QNM Eigenvalues --- p.232 / Chapter Appendix G. --- QNM Forward Problem 一 Transcendental Equation --- p.239 / Chapter Appendix H. --- Forward Problem - Calculation of Phase Shifts --- p.243 / Bibliography --- p.245

<p>The waveform of a radar signal affects the resolution in velocity and distance. The ambiguity function is used as an aid for analysing narrow band radar signals simultaneously in time and frequency. An analysing tool for wide band radar signals is missing.</p><p>This thesis describes a generalised ambiguity function to be utilised for study of wide band signals. Waveforms are further synthesised with help of the developed analysing tool. The aim is to start with a certain ambiguity function and find a waveform that reproduces the same ambiguity function.</p><p>Mathematical formulas are presented and implemented in Matlab to produce the wide band ambiguity function. Functions for developing waveforms by synthesis is also implemented.</p><p>It turns out that the Hermitian functions used as base functions do not preserve the orthogonality when implemented as wide band signals. The synthesis is not fully successful. Therefore an alternative method with numerical optimisation is used in an attempt to find an optimal waveform.</p>

The waveform of a radar signal affects the resolution in velocity and distance. The ambiguity function is used as an aid for analysing narrow band radar signals simultaneously in time and frequency. An analysing tool for wide band radar signals is missing. This thesis describes a generalised ambiguity function to be utilised for study of wide band signals. Waveforms are further synthesised with help of the developed analysing tool. The aim is to start with a certain ambiguity function and find a waveform that reproduces the same ambiguity function. Mathematical formulas are presented and implemented in Matlab to produce the wide band ambiguity function. Functions for developing waveforms by synthesis is also implemented. It turns out that the Hermitian functions used as base functions do not preserve the orthogonality when implemented as wide band signals. The synthesis is not fully successful. Therefore an alternative method with numerical optimisation is used in an attempt to find an optimal waveform.

Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (leaves [86]-88).

Thesis (MSc (Physical and Mathematical Analysis))--University of Stellenbosch, 2009. / In this thesis we aim to show that the Schr odinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian. After a brief introduction into non-Hermiticity, we will focus on solving the Schr odinger equation with a special class of non-Hermitian Hamiltonians, namely PT - symmetric Hamiltonians. PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing speci cally on obtaining the real and discrete energy spectrum. Various methods for solving this problematic Schr odinger equation will be considered. After starting with perturbation theory, we will move on to numerical methods. Three di erent categories of methods will be discussed. First there is the shooting method based on a Runge-Kutta solver. Next, we investigate various implementations of the spectral method. Finally, we will look at the Riccati-Pad e method, which is a numerical implemented analytical method. PT -symmetric potentials need to be solved along a contour in the complex plane. We will propose modi cations to the numerical methods to handle this. After solving the widely documented PT -symmetric Hamiltonian H = p2 􀀀(ix)N with these methods, we give a discussion and comparison of the obtained results. Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their in uence on the results.