Semiclassical asymptotics for the scattering amplitude in the presence of focal points at infinity [Elektronische Ressource] / von Horst Hohberger

Institut fur MathematikMathematische Physik: Semiklassik und AsymptotikSemiclassical asymptotics for thescattering amplitude in thepresence of focal points at in nityDissertationzur Erlangung des akademischen Grades“doctor rerum naturalium”(Dr. rer. nat.)in der Wissenschaftsdisziplin “Mathematische Physik”eingereicht an derMathematisch-Naturwissenschaftlichen Fakult atder Universit at PotsdamvonHorst HohbergerPotsdam, den 16. Oktober 2006ContentsIntroduction and statement of the main result 1Chapter 1. The scattering problem in classical phase space 51.1. The classical setting 51.2. Phase trajectories dened at in nity 61.3. The scattering manifold in euclidean phase space 18Chapter 2. The scattering manifold at in nity; caustics 232.1. The asymptotic manifold over the sphere 232.2. The Isozaki-Kitada phase functions 302.3. Generating functions 332.4. Lagrangian coordinates 36Chapter 3. The scattering amplitude 433.1. The representation formula 433.2. The Maslov Operator in extended phase space 463.3. The scattering amplitude as a Maslov operator on the asymptotic manifold 533.4. Perspectives and remarks 633.5. On caustics at in nity 64Appendix A. Some multi–index calculus 69Appendix B. Some properties of smooth maps 75Appendix C. Proof of Lemma 3.3.12 77Appendix D.