Eur. Phys. J. D (2015) 69: 5
https://doi.org/10.1140/epjd/e2014-40756-8

Regular Article

Logarithmic decays of unstable states

¹ Quantum Research Group, School of Chemistry and Physics, University of KwaZulu-Natal and National Institute for Theoretical Physics, KwaZulu-Natal, Westville Campus, 4000 Durban, South Africa
² Gruppo Nazionale per la Fisica Matematica (GNFM-INdAM), c/o Istituto Nazionale di Alta Matematica Francesco Severi Citta’ Universitaria, Piazza Aldo Moro 5, 00185 Roma, Italy

^a e-mail: filgi@libero.it

Received: 1 December 2013
Received in final form: 29 March 2014
Published online: 8 January 2015

Abstract

It is known that the survival amplitude of unstable quantum states deviates from exponential relaxations and exhibits decays that depend on the integral and analytic properties of the energy distribution density. In the same scenario, model independent dominant logarithmic decays t^−1−α₀log t of the survival amplitude are induced over long times by special conditions on the energy distribution density. While the instantaneous decay rate exhibits the dominant long time relaxation 1 /t, the instantaneous energy tends to the minimum value of the energy spectrum with the dominant logarithmic decay 1/(tlog ²t) over long times. Similar logarithmic relaxations have already been found in the dynamics of short range potential systems with even dimensional space or in the Weisskopf-Wigner model of spontaneous emission from a two-level atom. Here, logarithmic decays are obtained as a pure model independent quantum effect in general unstable states.