On the asymptotic behavior of the solutions of the Caldirola-Kanai equation

Abstract

We study in this Letter the asymptotic behavior, as t→+∞, of the solutions of the one-dimensional Caldirola-Kanai equation for a large class of potentials satisfying the condition V(x)→+∞ as |x|→∞. We show, first of all, that if I is a closed interval containing no critical points of V, then the probability P _t (t) of finding the particle inside I tends to zero as t→+∞. On the other hand, when I contains critical points of V in its interior, we prove that P _t (t) does not oscillate indefinitely, but tends to a limit as t→+∞. In particular, when the potential has only isolated critical points x ₁, ..., x _N our results imply that the probability density of the particle tends to $\sum\nolimits_{k = 1}^N {{\text{ }}c_k {\text{ }}\delta (x - x_k )}$ in the sense of distributions.