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UNIVERSITY OF BUCHAREST
FACULTY OF PHYSICS Guest
2024-11-21 20:33 |
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Conference: Bucharest University Faculty of Physics 2008 Meeting
Section: Theoretical Physics and Applied Mathematics Seminar
Title:
Non-exponential decay laws in perturbation
theory of near threshold eigenvalues
Authors:
Gheorghe Nenciu (1), Arne Jensen (2), Victor Dinu (3)
Affiliation:
(1)CAQP, Faculty of Physics, University of Bucharest and
Institute of Mathematics of the Romanian Academy
(2) Department of Mathematical Sciences, Aalborg University
(3) CAQP, Faculty of Physics, University of Bucharest
E-mail
Gheorghe.Nenciu@imar.ro
victordinu@yahoo.com
Keywords:
resonance, perturbation, decay, threshold, resolvent expansion, mathematical physics, Feshbach
Abstract:
The common wisdom predicts a quasi-exponential decay of resonances produced by perturbation of unstable bound states (exponential up to error terms wanishing with the strength of the perturbation). This is the generic result, when the resolvent of the unperturbed Hamiltonian, projected onto the subspace orthogonal to the considered bound state eigenfunction, is sufficiently smooth in a neighborhood of the unperturbed bound state energy. This may not be true near a threshold.
The problem of the decay law for threshold eigenvalues has been considered by us in previous studies under the condition that the shift in the energy due to the perturbation is sufficiently large, such that the resonance position is still “far” from the threshold. We proved that in this case the decay law is still (quasi)-exponential, but the Fermi golden rule (i.e. the width of the resonance) has to be modified.
In this study we remove that condition and allow arbitrary location of the resonance by tuning the unperturbed bound state energy across the threshold. Since this can be done experimentally for Feshbach resonances (with the aid of a magnetic field), our conditions on the Hamiltonians are modelled after those used in Feshbach resonances theory.
The main result is that in some energy windows, which depend upon the spectral properties of the unperturbed Hamiltonian at threshold, the decay law is definitely non-exponential for all times. We identify the time scales on which the survival probability varies significantly, as well as the decay law. Also we estimate the errors (vanishing as the perturbation strength parameter tends to zero) giving the first rigorous proof (to the best of our knowledge) of a non-exponential decay law in the perturbative framework.
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