Time asymptotics of e −ith(κ) for analytic matrices and analytic perturbation theory

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.