Long-time deviations from exponential decay for inverse-square potentials

Abstract

Quantal systems are predicted to show a changeover from exponential to a slower decay rate at very long times. Asymptotically most models predict power-law decay with integer exponents. However, the postexponential decay of a trapped particle from a potential can occur with a continuous range of power-law exponents. We show that this happens when the outer part of the potential is repulsive and decreases with the inverse square of the distance. We demonstrate a simple relation between the strength of the long-range tail and the power-law exponent. We also give explicit forms for the postexponential decay laws when the outer part of the potential is weakly attractive. These systems are amenable to experimental scrutiny.