On the nature of the quantum corrections in the case of stochastic motion of a nolinear oscillator

V. V. Sokolov

Teoretičeskaâ i matematičeskaâ fizika, Tome 61 (1984) no. 1, pp. 128-139

Résumé

It is known [1] that quantum effects leads to an exponentially rapid destruction of the stochastic phase trajectory of a nonlinear oscillator excited by a regular force exerted on it by an external source. In the present paper, the exponentially growing quantum corrections are completely summed. As a result, the semiclassical series for the quantum-mechanical expectation values is rearranged in such a way that it no longer contains the exponentially growing terms. The main term of the obtained series leads in the case of stochastic motion to the same dependence of the mean action of the oscillator on the time as in the classical case, and the first correction has the order $\hbar^2$. At the same time, the mean powers of the action contain already in the leading approximation corrections of order $\hbar$ that grow as powers with the time, though they do not change the asymptotic behavior as $t\rightarrow\infty$. The obtained results are not sensitive to the choice of the initial state of the oscillator.