by Ye Yuanling. / "August 2000." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (p. 82-85). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

In the limit we analyze the generators of families of reversible jump processes in the n-dimensional space associated with a class of symmetric non-local Dirichlet forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of certain eikonal equation. Fine results are sensitive to the rate functions being twice differentiable or just Lipschitz. Our estimates are similar to the semiclassical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice.

We study the integral equation related to the three and higher dimensional superradiance problem. Collective radiation phenomena has attracted the attention of many physicists and chemists since the pioneering work of R. H. Dicke in 1954. We first consider the three-dimensional superradiance problem and find a differential operator that commutes with the integral operator related to the problem. We find all the eigenfunctions of the differential operator and obtain a complete set of eigensolutions for the three-dimensional superradiance problem. Generalization of the three-dimensional superradiance integral equation is provided. A commuting differential operator is found for this generalized problem. For the three dimensional superradiance problem, an alternative set of complete eigenfunctions is also provided. The kernel for the superradiance problem when restricted to one-dimension is the same as appeared in the works of Slepian, Landau and Pollak. The uniqueness of the differential operator commuting with that kernel is indicated. Finally, a concentration problem for the signals which are bandlimited in disjoint frequency-intervals is considered. The problem is to determine which bandlimited signals lose the smallest fraction of their energy when restricted in a given time interval. A numerical algorithm for solution and convergence theorems are given. Orthogonality properties of analytically extended eigenfunctions over L2(−∞,∞) are also proved. Numerical computations are carried out in support of the theory.

This paper presents the investigation of the nonlinear eigenvalue problem describing free vibrations of plates with elastically attached masses. We study properties of eigenvalues and eigenfunctions and prove the existence theorem. Theoretical results are illustrated by numerical experiments.

Numerical methods have been used to predict the eigenperiods and eigenfunctions of the Earth’s Slichter modes, known as the Slichter triplets. In order to test the validity of our method, we have also computed the frequencies and displacement eigenfunctions of some of the inertial modes of the Earth’s fluid core. We use a Galerkin method to integrate the Three Potential Description (3PD) for a neutrally, stratified and rotating fluid core of a modified Preliminary Reference Earth Model (PREM). Moreover, the same mathematical tool is used for the computation of the frequencies and displacement amplitudes of the Slichter modes. In the Galerkin formulation of the 3PD, using the divergence theorem, we make use of the natural character of the boundary conditions to reduce the order of derivatives from second to first. To compute the frequencies of the Slichter modes, we solve simultaneously the equations of the inner core motion and the dynamics of the fluid core as described above. The results are compared to those in previous studies and it is shown that in the case of the inertial modes they agree well, which proves the validity of the approach. For the Slichter modes, however, it is shown that the results are significantly different from previous work for a similar Earth model. We have also plotted the displacement eigenfunctions for the motion of the fluid in the fluid core during the Slichter oscillations. It is shown that the pattern of motion is consistent with the motion of the inner core, which serves as a second test of the validity of our results. / x, 105 leaves : ill. ; 29 cm

Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1992. / Vita. Abstract. Includes bibliographical references (leaves 52-54). Also available via the Internet.

Thesis (Ph. D.)--University of California, Riverside, 2009. / Includes abstract. Includes bibliographical references (leaves 92-93). Issued in print and online. Available via ProQuest Digital Dissertations.

The completeness of eigenfunctions for linearized equations is critical for many applications, such as the study of stability of solitary waves. In this thesis, we work with the Nonlinear Schr{\"o}dinger (NLS) equation, associated with the Zakharov-Shabat spectral problem. Firstly, we construct a complete set of eigenfunctions for the spectral problem. Our method involves working with an adjoint spectral problem and deriving completeness and orthogonality relations between eigenfunctions and adjoint eigenfunctions. Furthermore, we prove completeness of squared eigenfunctions, which are used to represent solutions of the linearized NLS equation. For this, we find relations between the variation of potential and the variation of scattering data. Moreover, we show the connection between the squared eigenfunctions of the Zakharov-Shabat spectral problem and solutions of the linearized NLS equation. / Thesis / Master of Science (MSc)

This thesis consists of two parts. In the first part, we study a class of special functions Aik (y), k = 2, 4, 6, ··· generalising the classical Airy function Ai(y) to higher orders and in the second part, we apply expressions and properties of Ai4(y) to spectral problem of a specific operator. The first part is however motivated by latter part. We establish regularity properties of Aik (y) and particularly show that Aik (y) is smooth, bounded, and extends to the complex plane as an entire function, and obtain pointwise bounds on Aik (y) for all k. Some analytic properties of Aik (y) are also derived allowing one to express Aik (y) as a finite sum of certain generalised hypergeometric functions. We further obtain full asymptotic expansions of Aik (y) and their first derivative Ai'(y) both for y > 0 and for y < 0. Using these expansions, we derive expressions for the negative real zeroes of Aik (y) and Ai'(y). Using expressions and properties of Ai4(y), we extensively study spectral properties of a non-local operator H whose physical interpretation is the massless relativistic quartic anharmonic oscillator in one dimension. Various spectral results for H are derived including estimates of eigenvalues, spectral gaps and trace formula, and a Weyl-type asymptotic relation. We study asymptotic behaviour, analyticity, and uniform boundedness properties of the eigenfunctions Ψn(x) of H. The Fourier transforms of these eigenfunctions are expressed in two terms, one involving Ai4(y) and another term derived from Ai4(y) denoted by Āi4(y). By investigating the small effect generated by Āi4(y) this work shows that eigenvalues λn of H are exponentially close, with increasing n Ε N, to the negative real zeroes of Ai4(y) and those of its first derivative Ai'4(y) arranged in alternating and increasing order of magnitude. The eigenfunctions Ψ(x) are also shown to be exponentially well-approximated by the inverse Fourier transform of Ai4(|y| - λn) in its normalised form.

Le problème d’oscillation de fluides dans un conteneur est un problème classique d’hydrodynamique qui est etudié par des mathématiciens et ingénieurs depuis plus de 150 ans. Le présent travail est lié à l’étude de l’alternance des fonctions propres paires et impaires du problème de Steklov-Neumann pour les domaines à deux dimensions ayant une forme symétrique. On obtient des résultats sur la parité de deuxième et troisième fonctions propres d’un tel problème pour les trois premiers modes, dans le cas de domaines symétriques arbitraires. On étudie aussi la simplicité de deux premières valeurs propres non nulles d’un tel problème. Il existe nombre d’hypothèses voulant que pour le cas des domaines symétriques, toutes les valeurs propres sont simples. Il y a des résultats de Kozlov, Kuznetsov et Motygin [1] sur la simplicité de la première valeur propre non nulle obtenue pour les domaines satisfaisants la condition de John. Dans ce travail, il est montré que pour les domaines symétriques, la deuxième valeur propre non-nulle du problème de Steklov-Neumann est aussi simple. / The study of liquid sloshing in a container is a classical problem of hydrodynamics that has been actively investigated by mathematicians and engineers over the past 150 years. The present thesis is concerned with the properties of eigenfunctions of the two-dimensional sloshing problem on axially symmetric planar domains. Here the axis of symmetry is assumed to be orthogonal to the free surface of the fluid. In particular, we show that the second and the third eigenfunctions of such a problem are, respectively, odd and even with respect to the axial symmetry. There is a well-known conjecture that all eigenvalues of the two-dimensional sloshing problem are simple. Kozlov, Kuznetsov and Motygin [1] proved the simplicity of the first non-zero eigenvalue for domains satisfying the John's condition. In the thesis we show that for axially symmetric planar domains, the first two non-zero eigenvalues of the sloshing problem are simple.