Nonexponential decay laws in perturbation theory of near threshold eigenvalues

Abstract

We consider a two channel model of the formThe operator H op is assumed to have the properties of a Schrödinger operator in odd dimensions, with a threshold at zero. As the energy parameter E 0 is tuned past the threshold, we consider the survival probability | Ψ 0 , e −itHε Ψ 0 | 2 , where Ψ 0 is the eigenfunction corresponding to eigenvalue E 0 for ε = 0. We find non-exponential decay laws for ε small and E 0 close to zero, provided that the resolvent of H op is not at least Lipschitz continuous at the threshold zero.

Tosio Kato’s work on non-relativistic quantum mechanics: part 2

“…4; rather, as shown in Jensen-Nenciu [290], one typically has that it is O(|ξ − ξ j | 3/2 ). For related results, see Dinu et al [118,119].…”

Tosio Kato’s work on non-relativistic quantum mechanics, Part 2

“…4; rather, as shown in [277], one typically has that it is O(|ξ − ξ j | 3/2 ). For related results, see [115,116].…”

Perturbation of Near Threshold Eigenvalues: Crossover From Exponential to Non-Exponential Decay Laws

“…The other case, when the resonance position is very close to the threshold (in a neighborhood of the threshold, shrinking as ε → 0), has been considered in [8] for a two channel model Hamiltonian with the structure used in Feshbach resonance theory [21,30]. The main result is the proof at the rigorous level that for some energy ranges the decay law is definitely nonexponential.…”

“…The present paper is a continuation of [8]. The setting is the same, but we add two important things.…”

“…The contents of the paper is as follows. In Section 2 we recall from [8] the model Hamiltonian and its properties. Section 3 contains the guiding heuristics discussion, and a detailed description of the results.…”