Asymptotic spectral analysis and tunnelling for a class of difference operators [Elektronische Ressource] / von Elke Rosenberger

Institut fur MathematikMathematische Physik: Semiklassik und AsymptotikAsymptotic Spectral Analysis and Tunnellingfor a class of Di erence OperatorsDissertationzur Erlangung des akademischen Grades“doctor rerum naturalium”(Dr.rer.nat.)in der Wissenschaftsdisziplin “Mathematische Physik”eingereicht an derMathematisch-Naturwissenschaftlichen Fakultatder Universitat PotsdamvonElke RosenbergerPotsdam, den 8. April 2006ContentsChapter 1. Introduction 11.1. Denition of the Operator Class 11.2. General Strategy and Main Results 21.3. Classi cation and Motivation 41.4. Structure of this work 61.5. Open Questions related to this work 9Chapter 2. Stability of the spectrum 112.1. Notations and Preliminaries 112.2. Harmonic Approximation of the Spectrum of H 15ε2.3. Probabilistic Operator 32Chapter 3. Construction of asymptotic expansions 373.1. Hypothesis and motivation 373.2. Solution of the Eikonal Equation 393.3. Transformation of the variable and formal symbol spaces 413.4. Construction of asymptotic expansions 493.5. Constr of Asymptotic Expansions in x and ε 54Chapter 4. Finsler Distance associated to H 61ε4.1. Denition and Properties of Finsler Manifold and Finsler Metric 614.2. Finsler Function adapted to a hyperregular Hamiltonian 644.3. Geodesics as base integral curves of the associated vector eld 724.4. Application to H and the Eikonal (in)-equality 76εChapter 5. Weighted estimates for Dirichlet eigenfunctions 815.1.