Stochastic approach to generalized Schr{\"o}dinger equation: A method of eigenfunction expansion

Using a method of eigenfunction expansion, a stochastic equation is developed for the generalized Schr{\"o}dinger equation with random fluctuations. The wave field ${\psi}$ is expanded in terms of eigenfunctions: ${\psi} = \sum_{n} a_{n} (t) {\phi}_{n} (x)$, with ${\phi}_{n}$ being the eigenfunction that satisfies the eigenvalue equation $H_{0} {\phi}_{n} = {\lambda}_{n} {\phi}_{n}$, where $H_{0}$ is the reference "Hamiltonian" conventionally called "unperturbed" Hamiltonian. The Langevin equation is derived for the expansion coefficient $a_{n} (t)$, and it is converted to the Fokker--Planck (FP) equation for a set $\{ a_{n} \}$ under the assumption of the Gaussian white noise for the fluctuation. This procedure is carried out by a functional integral, in which the functional Jacobian plays a crucial role for determining the form of the FP equation. The analyses are given for the FP equation by adopting several approximate schemes.