On the asymptotic behavior of the solutions of the Caldirola-Kanai equation (Q1091542)

We study the asymptotic behavior, as \(t\to +\infty\), of the solutions of the one-dimensional Caldirola-Kanai equation for a large class of potentials satisfying the condition \(V(x)\to +\infty\) as \(| x| \to \infty\). We show, first of all, that if I is a closed interval containing no critical points of V, then the probability \(P_ I(t)\) of finding the particle inside I tends to zero as \(t\to +\infty\). On the other hand, when I contains critical points of V in its interior, we prove that \(P_ I(t)\) does not oscillate indefinitely, but tends to a limit as \(t\to +\infty\). 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 \(\sum^{N}_{k=1}c_ k\delta (x-x_ k)\) in the sense of distributions.