Dissertations / Theses on the topic 'Self-adjoint operators'

This thesis concerns how to compute upper and lower bounds for the eigenvalues of self-adjoint operators. We discuss two different methods: the so-called quadratic method and the Zimmermann-Mertins method. We know that the classical methods of computing the spectrum of a self-adjoint operator often lead to spurious eigenvalues in gaps between two parts of the essential spectrum. The methods to be examined have been studied recently in connection with the phenomenon of spectral pollution. In the first part of the thesis we show how to obtain enclosures of the eigenvalues in both the quadratic method and the Zimmermann-Mertins method. We examine the convergence properties of these methods for computing corresponding upper and lower bounds in the case of semi-definite self-adjoint operators with compact resolvent. In the second part of the thesis we find concrete asymptotic bounds for the size of the enclosure and study their optimality in the context of one-dimensional Schr¨odinger operators. The effectiveness of these methods is then illustrated by numerical experiments on the harmonic and the anharmonic oscillators. We compare these two methods, and establish which one is better suited in terms of accuracy and efficiency.

L'importance des opérateurs non auto-adjoints dans la physique moderne augmente chaque jour, car ils commencent à jouer un rôle plus important dans la mécanique quantique. Cependant, la signification de leur examen est beaucoup plus récente que l'intérêt pour l'examen des opérateurs auto-adjoints. Ainsi, étant donné que de nombreuses techniques auto-adjointes ne sont pas généralisées à ce contexte, il n’existe pas beaucoup de méthodes bien développées pour examiner leurs propriétés. Cette thèse vise à contribuer à combler cette lacune et démontre plusieurs modèles non auto-adjoints et les moyens de leur étude. Les sujets comprennent le pseudo-spectre comme un analogue approprié du spectre, un modèle d'une guide d'onde avec un gain et une perte équilibrés à la frontière et l'équation de Kramers-Fokker-Planck avec un potentiel à courte distance
The importance of non-self-adjoint operators in modern physics increases every day as they start to play more prominent role in Quantum mechanics. However, the significance of their examination is much more recent than the interest in the examination of their selfadjoint counterparts. Thus, since many selfadjoint techniques fail to be generalized to this context, there are not many well-developed methods for examining their properties. This thesis aims to contribute to filling this gap and demonstrates several non-self-adjoint models and the means of their study. The topics include pseudospectrum as a suitable analogue of the spectrum, a model of a quantum layer with balanced gain and loss at the boundary, and the Kramers-Fokker-Planck equation with a short-range potential

This thesis contains five chapters. The first two are devoted to the background which consists of integration, Fourier analysis, distributions and linear operators in Hilbert spaces. The third chapter is a generalization of a work done by Albrecht-Spain in 2000. We give a shorter proof of the main theorem they proved for bounded operators and we generalize it to unbounded operators. We give a counterexample that shows that the result fails to be true for another class of operators. We also say why it does not hold. In chapters four and five, the idea is the same, that is to find classes of unbounded, real-valued Vs for which  + V is self-adjoint on D(), where  is the wave operator. Throughout these two chapters we will see how different the Laplacian and the wave operator can be.

Let Hsub(0), Hsub(1) be Hilbert spaces and L : Hsub(0) -> Hsub(1) be a linear bounded operator with ||L|| ≤ 1. Then L*L is a bounded linear self-adjoint non-negative operator in the Hilbert space Hsub(0) and one can use the Neumann series ∑∞sub(v=0)(I - L*L)v L*f in order to study solvability of the operator equation Lu = f. In particular, applying this method to the ill-posed Cauchy problem for solutions to an elliptic system Pu = 0 of linear PDE's of order p with smooth coefficients we obtain solvability conditions and representation formulae for solutions of the problem in Hardy spaces whenever these solutions exist. For the Cauchy-Riemann system in C the summands of the Neumann series are iterations of the Cauchy type integral. We also obtain similar results 1) for the equation Pu = f in Sobolev spaces, 2) for the Dirichlet problem and 3) for the Neumann problem related to operator P*P if P is a homogeneous first order operator and its coefficients are constant. In these cases the representations involve sums of series whose terms are iterations of integro-differential operators, while the solvability conditions consist of convergence of the series together with trivial necessary conditions.

Thesis (Ph.D.)--Baylor University, 2009.
Subscript in abstract: n and n=0 in {Pn([alpha],[beta])(x)} [infinity] n=0, [mu] in (f,g)[mu], and R in [integral]Rfgd[mu]. Superscript in abstract: ([alpha],[beta]) and [infinity] in {Pn([alpha],[beta])(x)} [infinity] n=0. Includes bibliographical references (p. 115-119).

The convergence rates for the method of Weinstein and a variant method of Aronszajn known as "truncation including the remainder" are derived in terms of the containment gaps between exact and approximating subspaces, using analytical techniques that arise in part in the convergence analysis of finite element methods for differential eigenvalue problems. An example of a one dimensional Schrodinger operator with a potential is presented which arises in quantum mechanics. Examples using the recent eigenvector-free (EVF) method of Beattie and Goerisch are considered. Since the EVF method uses finite element trial functions as approximating vectors, it produces sparse and well-structured coefficient matrices. For these large-order sparse matrix eigenvalue problems, we adapt a spectral transformation Lanczos algorithm for finding a few wanted eigenvalues. For a few particular examples of vibration in beams and plates, convergence behavior is experimentally evaluated.
Ph. D.

We study several unbounded operators with view to extending von Neumann's theory of deficiency indices for single Hermitian operators with dense domain in Hilbert space. If the operators are non-commuting, the problems are difficult, but special cases may be understood with the use representation theory. We will further study the partial derivative operators in the coordinate directions on the L2 space on various covering surfaces of the punctured plane. The operators are defined on the common dense domain of C∞ functions with compact support, and they separately are essentially selfadjoint, but the unique selfadjoint extensions will be non-commuting. This problem is of a geometric flavor, and we study an index formulation for its solution. The applications include the study of vector fields, the theory of Dirichlet problems for second order partial differential operators (PDOs), Sturm-Liouville problems, H.Weyl's limit-point/limit-circle theory, Schrödinger equations, and more.

ix, 99 p.
We characterize the diagonals of four classes of self-adjoint operators on infinite dimensional Hilbert spaces. These results are motivated by the classical Schur-Horn theorem, which characterizes the diagonals of self-adjoint matrices on finite dimensional Hilbert spaces. In Chapters II and III we present some known results. First, we generalize the Schur-Horn theorem to finite rank operators. Next, we state Kadison's theorem, which gives a simple necessary and sufficient condition for a sequence to be the diagonal of a projection. We present a new constructive proof of the sufficiency direction of Kadison's theorem, which is referred to as the Carpenter's Theorem. Our first original Schur-Horn type theorem is presented in Chapter IV. We look at operators with three points in the spectrum and obtain a characterization of the diagonals analogous to Kadison's result. In the final two chapters we investigate a Schur-Horn type problem motivated by a problem in frame theory. In Chapter V we look at the connection between frames and diagonals of locally invertible operators. Finally, in Chapter VI we give a characterization of the diagonals of locally invertible operators, which in turn gives a characterization of the sequences which arise as the norms of frames with specified frame bounds. This dissertation includes previously published co-authored material.
Committee in charge: Marcin Bownik, Chair; N. Christopher Phillips, Member; Yuan Xu, Member; David Levin, Member; Dietrich Belitz, Outside Member

Orientador: Patricio Anibal Letelier Sotomayor
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Espaços-tempo classicamente singulares serão estudados de um ponto de vista quântico. A utilização da mecânica quântica será feita de duas maneiras. A primeira consiste em encontrar a função de onda do Universo, resolvendo a equação de Wheeler-DeWitt para as variáveis canônicas do espaço-tempo. A segunda consiste em acoplar conformemente campos escalares e spinoriais ao campo gravitacional, estudando o comportamento de pacotes de ondas neste espaço-tempo curvo
Abstract: Classically singular spacetimes will be studied from a quantum mechanical point of view. The use of quantum mechanics will be handled in two different ways. The first consists in finding the wave function of the universe by solving the Wheeler-DeWitt equation for the canonical variables of spacetime. The second is through the conformal coupling of scalar and spinorial fields with the gravitational field, where we will study the behavior of wave packets in this curved spacetime
Doutorado
Matematica Aplicada
Doutor em Matemática Aplicada

Dans cette thèse, nous nous intéressons aux propriétés spectrales des opérateurs non-auto-adjoints aléatoires. Nous allons considérer principalement les cas des petites perturbations aléatoires de deux types des opérateurs non-auto-adjoints suivants :1. une classe d’opérateurs non-auto-adjoints h-différentiels Ph, introduite par M. Hager [32],dans la limite semiclassique (h→0); 2. des grandes matrices de Jordan quand la dimension devient grande (N→∞). Dans le premier cas nous considérons l’opérateur Ph soumis à de petites perturbations aléatoires. De plus, nous imposons que la constante de couplage δ vérifie e (-1/Ch) ≤ δ ⩽ h(k), pour certaines constantes C, k > 0 choisies assez grandes. Soit ∑ l’adhérence de l’image du symbole principal de Ph. De précédents résultats par M. Hager [32], W. Bordeaux-Montrieux [4] et J. Sjöstrand [67] montrent que, pour le même opérateur, si l’on choisit δ ⪢ e(-1/Ch), alors la distribution des valeurs propres est donnée par une loi de Weyl jusqu’à une distance ⪢ (-h ln δ h) 2/3 du bord de ∑. Nous étudions la mesure d’intensité à un et à deux points de la mesure de comptage aléatoire des valeurs propres de l’opérateur perturbé. En outre, nous démontrons des formules h-asymptotiques pour les densités par rapport à la mesure de Lebesgue de ces mesures qui décrivent le comportement d’un seul et de deux points du spectre dans ∑. En étudiant la densité de la mesure d’intensité à un point, nous prouvons qu’il y a une loi de Weyl à l’intérieur du pseudospectre,une zone d’accumulation des valeurs propres dûe à un effet tunnel près du bord du pseudospectre suivi par une zone où la densité décroît rapidement. En étudiant la densité de la mesure d’intensité à deux points, nous prouvons que deux valeurs propres sont répulsives à distance courte et indépendantes à grande distance à l’intérieur de ∑. Dans le deuxième cas, nous considérons des grands blocs de Jordan soumis à des petites perturbations aléatoires gaussiennes. Un résultat de E.B. Davies et M. Hager [16] montre que lorsque la dimension de la matrice devient grande, alors avec probabilité proche de 1, la plupart des valeurs propres sont proches d’un cercle. De plus, ils donnent une majoration logarithmique du nombre de valeurs propres à l’intérieur de ce cercle. Nous étudions la répartition moyenne des valeurs propres à l’intérieur de ce cercle et nous en donnons une description asymptotique précise. En outre, nous démontrons que le terme principal de la densité est donné par la densité par rapport à la mesure de Lebesgue de la forme volume induite par la métrique de Poincaré sur la disque D(0, 1)
In this thesis we are interested in the spectral properties of random non-self-adjoint operators. Weare going to consider primarily the case of small random perturbations of the following two types of operators: 1. a class of non-self-adjoint h-differential operators Ph, introduced by M. Hager [32], in the semiclassical limit (h→0); 2. large Jordan block matrices as the dimension of the matrix gets large (N→∞). In case 1 we are going to consider the operator Ph subject to small Gaussian random perturbations. We let the perturbation coupling constant δ be e (-1/Ch) ≤ δ ⩽ h(k), for constants C, k > 0 suitably large. Let ∑ be the closure of the range of the principal symbol. Previous results on the same model by M. Hager [32], W. Bordeaux-Montrieux [4] and J. Sjöstrand [67] show that if δ ⪢ e(-1/Ch) there is, with a probability close to 1, a Weyl law for the eigenvalues in the interior of the pseudospectrumup to a distance ⪢ (-h ln δ h) 2/3 to the boundary of ∑. We will study the one- and two-point intensity measure of the random point process of eigenvalues of the randomly perturbed operator and prove h-asymptotic formulae for the respective Lebesgue densities describing the one- and two-point behavior of the eigenvalues in ∑. Using the density of the one-point intensity measure, we will give a complete description of the average eigenvalue density in ∑ describing as well the behavior of the eigenvalues at the pseudospectral boundary. We will show that there are three distinct regions of different spectral behavior in ∑. The interior of the of the pseudospectrum is solely governed by a Weyl law, close to its boundary there is a strong spectral accumulation given by a tunneling effect followed by a region where the density decays rapidly. Using the h-asymptotic formula for density of the two-point intensity measure we will show that two eigenvalues of randomly perturbed operator in the interior of ∑ exhibit close range repulsion and long range decoupling. In case 2 we will consider large Jordan block matrices subject to small Gaussian random perturbations. A result by E.B. Davies and M. Hager [16] shows that as the dimension of the matrix gets large, with probability close to 1, most of the eigenvalues are close to a circle. They, however, only state a logarithmic upper bound on the number of eigenvalues in the interior of that circle. We study the expected eigenvalue density of the perturbed Jordan block in the interior of thatcircle and give a precise asymptotic description. Furthermore, we show that the leading contribution of the density is given by the Lebesgue density of the volume form induced by the Poincarémetric on the disc D(0, 1)

Orientador: Patricio Anibal Letelier Sotomayor
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Espaços-tempos classicamente singulares são estudados utilizando-se partículas quânticas (ao invés de clássicas) obedecendo as equações de Klein-Gordon e Dirac, a fim de determinar se estes espaços permanecem singulares do ponto de vista quântico. Primeiramente é apresentada uma revisão do ferramental matemático necessário para o estudo de singularidades quânticas, cujo principal resultado utilizado é a teoria de índices deficientes devido a von Neumann. No apêndice A é apresentado um primeiro estudo sobre singularidades quânticas em espaços-tempos com defeitos topológicos numa superfície 2-dimensional (paredes cósmicas), em especial superfícies esféricas e cilíndricas. Estes espaços continuam singulares nesta teoria e todas as informações extras (que em mecânica quântica se apresentam sob a forma de condições de contorno) necessárias para se remover a singularidade são encontradas. No apêndice B, é estudado um espaço-tempo 2+1 dimensional com curvatura negativa constante. É mostrado que este espaço permanece singular quando visto pela mecânica quântica e as condições de contorno possíveis são encontradas utilizando-se resultados obtidos no caso plano
Abstract: Classical singular spacetimes are studied using quantum particles (instead of classical ones) obeying Klein-Gordon and Dirac equations, to determine if these spacetimes remain singular in the view of quantum mechanics. First we give a review of the mathematical framework necessary to study quantum singularities, wich the main result to be used later is von Neumann¿s theory of deficient indices. In appendix A, a first work on quantum singularities in spacetimes with topological defects on a 2-dimensional hypersurface (cosmic walls), specifically spherical and cylindrical surfaces, is presented. These spacetimes remain singular in this theory and all extra informations (which in quantum mechanics correspond to boundary conditions) necessary to remove the naked singularity are found. In Apendix B, a 2+1 dimensional spacetime with constant negative curvature is studied. It is shown that this spacetime remains quantum mechanically singular and all possible boundary conditions are found using results obtained in plane case
Mestrado
Relatividade Geral/Gravitação Quantica
Mestre em Matemática Aplicada

En application, on souhaite générer des nombres aléatoires avec une loi précise (méthode de Monte Carlo par chaines de Markov - MCMC (Markov Chaine Monte Carlo)). La méthode consiste à trouver une diffusion qui a la loi invariante souhaitée et à montrer la convergence de cette diffusion vers son équilibre avec une vitesse exponentielle. L’exposant de cette convergence est le trou spectral du générateur. Il a été montré par Chii-Ruey Hwang, Shu-Yin Hwang-Ma, et Shuenn-Jyi Sheu qu’on peut agrandir le trou spectral, en rajoutant un terme non-symétrique au générateur auto-adjoint (souvent utilisé en MCMC). Ceci correspond à passer d’une diffusion réversible (en detailed balance) à une diffusion non réversible. Un moyen de construire une diffusion non-réversible avec la même mesure invariante est de rajouter un flot incompressible à la dynamique de la diffusion réversible.Dans cette thèse, nous étudions le comportement de la diffusion lorsqu’on accélère le flot sous-jacent en multipliant le champ des vecteurs qui le décrit par une grande constante. P. Constantin, A.Kisekev, L.Ryzhik et A.Zlatoš (2008) ont montré que si le flot était faiblement mélangeant alors l’accélération du flot suffisait pour faire converger la diffusion vers son équilibre en un temps fini. Dans ce travail, on explicite la vitesse de ce phénomène sous une condition de corrélation du flot. L’article de B. Franke, C.-R.Hwang, H.-M. Pai et S.-J. Sheu (2010) donne l’expression asymptotique du trou spectral lorsque le flot sous-jacent est accéléré vers l’infini. Ici aussi, on s’intéresse à la vitesse avec laquelle le phénomène se manifeste. Dans un premier temps, nous étudions le cas particulier d’une diffusion du type Ornstein-Uhlenbeck qui est perturbée par un flot préservant la mesure gaussienne. Dans ce cas, grâce à un résultat de G. Metafune, D. Pallara et E. Priola (2002), nous pouvons réduire l’étude du spectre du générateur à des valeurs propres d’une famille de matrices. Nous étudions ce problème avec des méthodes de développement limité des valeurs propres. Ce problème est résolu explicitement dans cette thèse et nous donnons aussi une borne pour le rayon de convergence du développement. Nous généralisons ensuite cette méthode dans le cas d’une diffusion générale de façon formelle. Ces résultats peuvent être utiles pour avoir une première idée sur les vitesses de convergence du trou spectral décrites dans l’article de Franke et al. (2010)
In application, we would like to generate random numbers with a precise law MCMC (Markov Chaine Monte Carlo). The method consists in finding a diffusion which has the desired invariant law and in showing the convergence of this diffusion towards its equilibrium with an exponential rate. The exponent of this convergence is the spectral gap of the generator. It was shown by C.-R. Hwang, S.-Y. Hwang-Ma and S.-J. Sheu that the spectral gap can grow up by adding a non-symmetric term to the self-adjoint generator.This corresponds to passing from a reversible diffusion to a non-reversible diffusion. A means of constructing a non-reversible diffusion with the same invariant measure is to add an incompressible flow to the dynamics of the reversible diffusion.In this thesis, we study the behavior of diffusion when the flow is accelerated by multiplying the field of the vectors which describes it by a large constant. In 2008, P. Constantin, A. Kisekev, L. Ryzhik and A. Zlatoˇs have shown that if the flow was weakly mixing then the acceleration of the flow was sufficient to converge the diffusion towards its equilibrium after finite time. In this work, the speed of this phenomenon is explained under a condition of correlation of the flow. The article by B. Franke, C.-R.Hwang, H.-M. Pai and S.-J.Sheu (2010) gives the asymptotic expression of the spectral gap when the large constant goes to infinity. Here we are also interested in the speed with which the phenomenon manifests itself. First, we study the special case of an Ornstein-Uhlenbeck diffusion which is perturbed by a flow preserving the Gaussian measure. In this case, thanks to a result of G. Metafune, D. Pallara and E. Priola (2002), we can reduce the study of the generator spectrum to eigenvalues of a family of matrices. We study this problem with methods of limited development of eigenvalues. This problem is solved explicitly in this thesis and we also give a boundary for the convergence radius of the development. We then generalize this method in the case of a general diffusion in a formal way. These results may be useful to have a first idea on the speeds of convergence of the spectral gap described in the article by Franke et al. (2010)

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Financiadora de Estudos e Projetos
In this work we study the Hamiltonian of the hydrogen atom in 1, 2 and 3 dimensions. Especifically, it is defined as a self-adjoint operator in the Hilbert space L2(Rn), n = 1, 2, 3. Nevertheless, the main goal is to study the hydrogen atom 1-D. Particularly, for this is model we address some problens related to the singularity of the Coulomb potential.
Neste trabalho vamos estudar o Hamiltoniano do átomo de hidrogênio em 1, 2 e 3 dimensões. Especificamente, queremos defini-lo como um operador auto-adjunto no espaço de Hilbert L2(Rn), n = 1, 2, 3. No entanto, o principal objetivo é estudar o átomo de hidrogênio 1-D. Em particular, para este modelo, abordaremos algumas questões relacionadas à singularidade do potencial de Coulomb −1/|x|.

Dans cette thèse nous considérons l’équation des ondes amorties vectorielle sur une variété riemannienne compacte, lisse et sans bord. L’amortisseur est ici une fonction lisse allant de la variété dans l’espace des matrices hermitiennes de taille n. Les solutions de cette équation sont donc à valeurs vectorielles. Nous commençons dans un premier temps par calculer le meilleur taux de décroissance exponentiel de l’énergie en fonction du terme d’amortissement. Ceci nous permet d’obtenir une condition nécessaire et suffisante la stabilisation forte de l’équation des ondes amorties vectorielle. Nous mettons aussi en évidence l’apparition d’un phénomène de sur-amortissement haute fréquence qui n’existait pas dans le cas scalaire. Dans un second temps nous nous intéressons à la répartition asymptotique des fréquences propres de l’équation des ondes amorties vectorielle. Nous démontrons que, à un sous ensemble de densité nulle près, l’ensemble des fréquences propres est contenu dans une bande parallèle à l’axe imaginaire. La largeur de cette bande est déterminée par les exposants de Lyapunov d’un système dynamique défini à partir du coefficient d’amortissement
In this thesis we are considering the vectorial damped wave equation on a compact and smooth Riemannian manifold without boundary. The damping term is a smooth function from the manifold to the space of Hermitian matrices of size n. The solutions of this équation are thus vectorial. We start by computing the best exponential energy decay rate of the solutions in terms of the damping term. This allows us to deduce a sufficient and necessary condition for strong stabilization of the vectorial damped wave equation. We also show the appearance of a new phenomenon of high-frequency overdamping that did not exists in the scalar case. In the second half of the thesis we look at the asymptotic distribution of eigenfrequencies of the vectorial damped wave equation. Were show that, up to a null density subset, all the eigenfrequencies are in a strip parallel to the imaginary axis. The width of this strip is determined by the Lyapunov exponents of a dynamical system defined from the damping term

Cette thèse s'articule autour d'une notion spectrale assez récente, appelée le spectre étendu des opérateurs. Dans la première partie nous fournissons des propriétés générales du spectre étendu d'un opérateur dans certains cas particuliers, tels que le cas de dimension finie et celui des opérateurs inversibles. Nous nous intéressons dans la deuxième partie à l'étude du spectre étendu de l'opérateur shift tronqué Su. En particulier, nous donnons une description complète des vecteurs propres étendus associes à chaque valeur propre étendue de Sb, ou b est un produit de Blaschke quelconque. Dans la troisième partie nous décrirons complètement le spectre étendu et les sous espaces propres étendus d'une classe d'opérateurs très importante : celle des opérateurs normaux. Nous commençons d'abord par la classe des opérateurs qui sont produits d'un opérateur positif par un autoadjoint. Ensuite, nous utilisons le théorème de Fuglede-Putnam pour déduire une description complète des valeurs et des vecteurs propres étendus des opérateurs normaux, en fonction de leur mesure spectrale. Dans la dernière partie, nous appliquons nos résultats des trois premières parties sur des exemples concrets. En particulier, nous traitons= le problème des sous espaces propres étendus des opérateurs définis dans un espace de dimension finie. Ensuite, nous montrons l'existence d'un opérateur compact quasinilpotent dont le spectre étendu est réduit au singleton {1}. Enfin, nous traitons deux opérateurs de Cesaro très importants dans les applications
This thesis is based on a relatively new spectral notion, called extended spectrum of operators. In the first part, we provide general properties of extended spectrum of an operator in some special cases, such as the case of finite dimension and the case of invertible operator. We focused in the second part on characterizing the extended spectrum of truncated shift operator Su. In particular, we give a complete description of the extended eigenvectors associated to each extended eigenvalue of Sb, where b is a Blaschke product. In the third part, we describe the extended spectrum and the extended eigenvectors of a very important class of operators , that is the normal operators. We first start by describing these last sets for the product of a positive and a self-adjoint operator which are both injective. After, we use the Fuglede-Putnam theorem to describe the same sets for normal operators, in terms of their spectral measure. In the last part, we apply our results from the last three parts on concrete examples. In particular, we address the problem of extended eigenvectors of operators defined in a finite dimension space. Next, we show the existence of a quasinilpotent compact operator whose extended spectrum is reduced to {1}. Finally, we study two Cesaro operators which are very important in applications

Dans les chapitres 2 et 3 de cette these, on s'interesse aux sous-varietes de l'espace hyperbolique dont la courbure moyenne est constante et strictement inferieure a un. Le premier resultat qu'on obtient concerne l'operateur de stabilite. Dans notre cas, cet operateur est essentiellement auto-adjoint, et on connait un minorant positif de son spectre essentiel. On montre que le nombre de valeurs propres inferieures a ce minorant est fini, et on en obtient un majorant qui fait intervenir la courbure totale. Ce faisant, on obtient un majorant de l'indice de morse de sous-variete. Un des points importants de la preuve est de controler le noyau de la chaleur de la sous-variete. On obtient ce controle en montrant qu'on a sur la sous-variete des inegalites isoperimetriques. Le second resultat porte sur la compactification. On etend aux sous-varietes a courbure moyenne constantes un resultat de g. De oliveira pour les sous-varietes minimales: on montre que la sous-variete est diffeomorphe a l'interieur d'une variete compact a bord, et que l'immersion s'etend continument au bord en une application a valeurs dans le compactifie de l'espace hyperbolique. Dans le chapitre 4, on etudie les surfaces de revolution a courbure moyenne constante dans l'espace hyperbolique. On obtient une construction cinematique de leurs meridiennes analogue a celle donne par c. Delaunay dans l'espace euclidien. Les courbes a faire rouler ont des proprietes focales similaires a celles des coniques euclidiennes. On trouve les analogues hyperboliques des ellipses, des hyperboles ainsi qu'une surprenante famille de paraboles

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Master of Science. Johannesburg, March 2015.
The primary purpose of this study is to investigate the asymptotic distribution of the eigenvalues of self-adjoint second order di erential operators. We rst analyse the problem where the functions g and h are equal to zero. To improve on the terms of the eigenvalue problem for g; h = 0, we consider the eigenvalue problem for general functions g and h. Here we calculate explicitly the rst four terms of the eigenvalue asymptotics problem.

Sampling theory is an active field of research that spans a variety of disciplines from communication engineering to pure mathematics. Sampling theory provides the crucial connection between continuous and discrete representations of information that enables one store continuous signals as discrete, digital data with minimal error. It is this connection that allows communication engineers to realize many of our modern digital technologies including cell phones and compact disc players. This thesis focuses on certain non-Fourier generalizations of sampling theory and their applications. In particular, non-Fourier analogues of bandlimited functions and extensions of sampling theory to functions on curved manifolds are studied. New results in bandlimited function theory, sampling theory on curved manifolds, and the theory of self-adjoint extensions of symmetric operators are presented. Besides being of mathematical interest in itself, the research contained in this thesis has applications to quantum physics on curved space and could potentially lead to more efficient information storage methods in communication engineering.

The eigenvalue problem y(4)(¸; x) ¡ (gy0)0(¸; x) = ¸2y(¸; x) with boundary conditions y(¸; 0) = 0; y00(¸; 0) = 0; y(¸; a) = 0; y00(¸; a) + i®¸y0(¸; a) = 0; where g 2 C1[0; a] is a real valued function and ® > 0, has an operator pencil L(¸) = ¸2 ¡ i®¸K ¡ A realization with self-adjoint operators A, M and K. It was shown that the spectrum for the above boundary eigenvalue problem is located in the upper-half plane and on the imaginary axis. This is due to the fact that A, M and K are self-adjoint. We consider the eigenvalue problem y(4)(¸; x) ¡ (gy0)0(¸; x) = ¸2y(¸; x) with more general ¸-dependent separated boundary conditions Bj(¸)y = 0 for j = 1; ¢ ¢ ¢ ; 4 where Bj(¸)y = y[pj ](aj) or Bj(¸)y = y[pj ](aj) + i²j®¸y[qj ](aj), aj = 0 for j = 1; 2 and aj = a for j = 3; 4, ® > 0, ²j = ¡1 or ²j = 1. We assume that at least one of the B1(¸)y = 0, B2(¸)y = 0, B3(¸)y = 0, B4(¸)y = 0 is of the form y[p](0)+i²®¸y[q](0) = 0 or y[p](a)+i²®¸y[q](a) = 0 and we investigate classes of boundary conditions for which the corresponding operator A is self-adjoint.

We consider on the interval [0; a], rstly fourth-order di erential operators with eigenvalue parameter dependent boundary conditions and secondly a sixth-order di erential operator with eigenvalue parameter dependent boundary conditions. We associate to each of these problems a quadratic operator pencil with self-adjoint operators. We investigate the spectral proprieties of these problems, the location of the eigenvalues and we explicitly derive the rst four terms of the eigenvalue asymptotics.

Sampling theory studies the equivalence between continuous and discrete representations of information. This equivalence is ubiquitously used in communication engineering and signal processing. For example, it allows engineers to store continuous signals as discrete data on digital media. The classical sampling theorem, also known as the theorem of Whittaker-Shannon-Kotel'nikov, enables one to perfectly and stably reconstruct continuous signals with a constant bandwidth from their discrete samples at a constant Nyquist rate. The Nyquist rate depends on the bandwidth of the signals, namely, the frequency upper bound. Intuitively, a signal's `information density' and `effective bandwidth' should vary in time. Adjusting the sampling rate accordingly should improve the sampling efficiency and information storage. While this old idea has been pursued in numerous publications, fundamental problems have remained: How can a reliable concept of time-varying bandwidth been defined? How can samples taken at a time-varying Nyquist rate lead to perfect and stable reconstruction of the continuous signals? This thesis develops a new non-Fourier generalized sampling theory which takes samples only as often as necessary at a time-varying Nyquist rate and maintains the ability to perfectly reconstruct the signals. The resulting Nyquist rate is the critical sampling rate below which there is insufficient information to reconstruct the signal and above which there is redundancy in the stored samples. It is also optimal for the stability of reconstruction. To this end, following work by A. Kempf, the sampling points at a Nyquist rate are identified as the eigenvalues of self-adjoint extensions of a simple symmetric operator with deficiency indices (1,1). The thesis then develops and in a sense completes this theory. In particular, the thesis introduces and studies filtering, and yields key results on the stability and optimality of this new method. While these new results should greatly help in making time-variable sampling methods applicable in practice, the thesis also presents a range of new purely mathematical results. For example, the thesis presents new results that show how to explicitly calculate the eigenvalues of the complete set of self-adjoint extensions of such a symmetric operator in the Hilbert space. This result is of interest in the field of functional analysis where it advances von Neumann's theory of self-adjoint extensions.