Using a method of eigenfunction expansion, a stochastic equation is developed for the generalized Schrödinger equation with random fluctuations. The wave field $\psi$ is expanded in terms of eigenfunctions: $\psi ={\sum }_n{a}_n\left(t\right){\varphi }_n\left(x\right)$, with ${\varphi }_n$ being the eigenfunction that satisfies the eigenvalue equation $H_0{\varphi }_n={\lambda }_n{\varphi }_n$, where $H_0$ is the reference “Hamiltonian” conventionally called the “unperturbed” Hamiltonian. The Langevin equation is derived for the expansion coefficient $a_n\left(t\right)$, and it is converted to the Fokker-Planck (FP) equation for a set $\left\{a_n\right\}$ under the assumption of Gaussian white noise for the fluctuation. This procedure is carried out by a functional integral, in which the functional Jacobian plays a crucial role in determining the form of the FP equation. The analyses are given for the FP equation by adopting several approximate schemes.

• Received 6 January 2015

DOI:https://doi.org/10.1103/PhysRevE.91.052146