On the asymptotic behavior of the solutions of the Caldirola-Kanai equation |
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| Authors: | Ph De Smedt A González López |
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| Institution: | (1) Department of Mathematics, Rutgers University, 08903 New Brunswick, NJ, USA;(2) Physics Department, Princeton University, 08544 Prineton, NZ, USA;(3) Present address: Departamento de Métodos Matemáticos de la Física, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain |
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| 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
in the sense of distributions.Supported by Fulbright-MEC grant 85-07391. |
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| Keywords: | |
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