Affiliations: Institut für Mathematik, Universität Potsdam, Potsdam, Germany | Department of Mathematics, University of Virginia, Charlottesville, VA, USA
Note: [] Corresponding author: J. Rama, Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany. E-mail: [email protected]
Abstract: In quantum mechanics the temporal decay of certain resonance states is associated with an effective time evolution e−ith(κ), where h(·) is an analytic family of non-self-adjoint matrices. In general the corresponding resonance states do not decay exponentially in time. Using analytic perturbation theory, we derive asymptotic expansions for e−ith(κ), simultaneously in the limits κ→0 and t→∞, where the corrections with respect to pure exponential decay have uniform bounds in one complex variable κ2t. In the Appendix we briefly review analytic perturbation theory, replacing the classical reference to the 1920 book of Knopp [Funktionentheorie II, Anwendungen und Weiterführung der allgemeinen Theorie, Sammlung Göschen, Vereinigung wissenschaftlicher Verleger Walter de Gruyter, 1920] and its terminology by standard modern references. This might be of independent interest.
Keywords: resonances, exponential decay, long-time corrections, Fermi golden rule, analytic perturbation theory
