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Journal of Mathematical Physics

One more theorem on the short-time regeneration rate

Pavel Exner

Published Nov 1, 1989• 22 citations
Publisher

Abstract

Consider an unstable quantum system that has been found undecayed at an instant s and denote by R(t,s) the rate of its regeneration into an original (undecayed) state at a later instant t+s. It is proved that the reduced evolution is a semigroup, i.e., there is no regeneration at all, provided R(t,s)1/2 can be estimated by a sufficiently regular function that is nondecreasing in s and has zero derivative with respect to t at t=0 for every s. This generalizes the theorem of Misra and Sinha [Helv. Phys. Acta 45, 619 (1972)] in a different direction than in a recent paper by Nishioka [J. Math. Phys. 29, 1860 (1988)].

DOI
10.1063/1.528536

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Consider an unstable quantum system that has been found undecayed at an instant s and denote by R(t,s) the rate of its regeneration into an original (undecayed) state at a later instant t+s. It is proved that the reduced evolution is a semigroup, i.e., there is no regeneration at all, provided R(t,s)1/2 can be estimated by a sufficiently regular function that is nondecreasing in s and has zero derivative with respect to t at t=0 for every s. This generalizes the theorem of Misra and Sinha [Helv. Phys. Acta 45, 619 (1972)] in a different direction than in a recent paper by Nishioka [J. Math. Phys. 29, 1860 (1988)].
There is given a generalization of the theorem of Misra and Sinha [Helv. Phys. Acta 50, 99 (1977)], which shows that in the decay process the regeneration of the undecayed unstable states from the decay products cannot proceed too slowly at arbitrarily short time and, correspondingly, the deviation from exponential decay at arbitrarily short time cannot be too small either. The generalization presented here is shown to be valid in the region where the theorem of Misra and Sinha cannot be applied.
The authors study the regeneration contribution to the decay law in Wigner-Weisskopf theory for times less than and up to the golden rule time. A power series expansion for the regeneration term and the part of the product of the amplitudes which has the semigroup property is carried out in second-order perturbation theory, the same order to which the Wigner-Weisskopf calculation is carried out in their estimate of the line widths in atomic decay. They show that the regeneration contribution as a smaller leading behaviour in t than the amplitudes at times of the order of the golden rule time, thus accounting for an approximate semigroup behaviour, on this scale, within the framework of the Wigner-Weisskopf theory. For very short times, the estimates of Misra and Sinha (1977) are obtained.