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

Les équations différentielles constituent un important outil pour l'étude des variétés algébriques et analytiques, sur les nombres complexes et $p$-adiques. Dans le cas $p$-adique, elles présentent des phénomènes qui n'apparaissent pas dans le cas complexe. En effet, le rayon de convergence des solutions d'une équation différentielle linéaire peut être fini, et cela même en l'absence des pôles.La connaissance de ce rayon permet d’obtenir de nombreuses informations intéressantes sur l’équation. Plus précisément, depuis les travaux de F. Baldassarri, on sait associer un rayon de convergence à tout point d’une courbe p-adique au sens de Berkovich munie d’une connexion. Des travaux récents de F. Baldassarri, K. Kedlaya, J. Poineauet A. Pulita ont révélé que ce rayon se comporte de manière très contrainte. Afin de pousser l'étude, on introduit un objet géométrique qui raffine ce rayon, le spectre au sens de Berekovich d'une équation différentielle.Dans ce mémoire de thèse, nous définissons le spectre d'un module différentiel et donnons ses premières propriétés. Nous calculons aussi les spectres de quelques classes de modules différentiels: modules différentiels d'une équations différentielles à coefficients constants, modules différentiels singuliers réguliers et enfin modules différentiels sur un corps des séries de Laurent. / Differential equations constitute an important tool for theinvestigation of algebraic and analytic varieties, over thecomplex and the $p$-adic numbers. In the $p$-adic setting, theypresent phenomena that do not appear in the complex case. Indeed, theradius of convergence of the solutions of a linear differential equation,even without presence of poles.The knowledge of that radius permits to obtain several interestinginformations about the equation. More precisely, since the works ofF. Baldassarri, we know how to associate a radius of convergece to allpoint of a p-adic curve in the sense of Berkovich endowed with aconnexion. Recent works of F. Baldassarri, K.S. Kedlaya, J. Poineau, etA. Pulita have proved that this radius behave in a very controlledway. The radius of convergence can be refined using subsidiary radii,that are known to have similar properties. In order to push forward the study, we introduce a geometric object that refine this radius, thespectrum in the sense of Berkovich of a differential equation.In the present thesis, we define the spectrum of a differentialequation and provide its first properties. We also compute the spectraof some classes of differential modules: differential modules ofa differential équation with constant coefficients, singular regulardifferential modules and at last differential modules over the field ofLaurent power series.

Lo Ting Shek = 從開放系統頻譜或散射相移到逆有限支合集勢函數的研究 / 盧庭碩. / "April 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 144-146). / 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. / Lo Ting Shek = Cong kai fang xi tong pin pu huo san she xiang yi dao ni you xian zhi he ji shi han shu de yan jiu / Lu Tingshuo.

On compact Riemann surfaces, the Laplacian $\Delta$ has a discrete, non-negative spectrum of eigenvalues $\{\lambda_{i}\}$ of finite multiplicity. The spectrum is intrinsically linked to the geometry of the surface. In this work, we consider surfaces of constant negative curvature with a large symmetry group. It is not possible to explicitly calculate the eigenvalues for surfaces in this class, so we combine group theoretic and analytical methods to derive results about the spectrum. In particular, we focus on the Bolza surface and the Klein quartic. These have the highest order symmetry groups among compact Riemann surfaces of genera 2 and 3 respectively. The full automorphism group of the Bolza surface is isomorphic to $\mathrm{GL}_{2}(\mathbb{Z}_{3})\rtimes\mathbb{Z}_{2}. We analyze the irreducible representations of this group and prove that the multiplicity of $\lambda_{1}$ is 3, building on the work of Jenni, and identify the irreducible representation that corresponds to this eigenspace. This proof relies on a certain conjecture, for which we give substantial numerical evidence and a hopeful method for proving. We go on to show that $\lambda_{2}$ has multiplicity 4.

While the spectral theory of compact operators is known to many, knowledge regarding the relationship between eigenvalues and approximation numbers might be less known. By examining these numbers in tandem, one may develop a link between eigenvalues and l^p spaces. In this paper, we develop the background of this connection with in-depth examples.

Some eigenvalue inequalities for Klein-Gordon operators and fractional Laplacians restricted to a bounded domain are proved. Such operators became very popular recently as they arise in many problems ranging from mathematical finance to crystal dislocations, especially relativistic quantum mechanics and symmetric stable stochastic processes. Many of the results obtained here are concerned with finding bounds for some functions of the spectrum of these operators. The subject, which is well developed for the Laplacian, is examined from the spectral theory perspective through some of the tools used to prove analogous results for the Laplacian. This work highlights some important results, sparking interest in constructing a similar theory for Klein-Gordon operators. For instance, the Weyl asymptotics and semiclassical bounds for the Klein-Gordon operator are developed. As a result, a Berezin-Li-Yau type inequality is derived and an improvement of the bound is proved in a separate chapter. Other results involving some universal bounds for the Klein-Gordon Hamiltonian with an external interaction are also obtained.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 128-134). Also available on the Internet.

Tez (Yüksek Lisans) - Süleyman Demirel Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Anabilim Dalı, 2008. / Kaynakça var.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 128-134). Also available on the Internet.

Thesis (M. Phil.)--University of Hong Kong, 2002. / Includes bibliographical references (leaves.