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

We study in this Letter the asymptotic behavior, as t→ + ∞, of the solutions of the onedimensional Caldirola-Kanai equation for a large class of pontentials satisfying the condition V(x)→ + ∞ as vertical strokexvertical stroke→∞. We show, first of all, that if I is a closed interval containing no critical points of V, then the probability PI(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 Pl(t) does not oscillate indefinitely, but tends to a limit as t→ + ∞. In particular, when the potential has only isolated critical points x1,...,xN our results imply that the probability density of the particle tends to ΣNk=1ckδ(x-xk) in the sense of distributions. (orig.)