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Stochastic approach to generalized Schr{ö}dinger equation: A method of eigenfunction expansion

Using a method of eigenfunction expansion, a stochastic equation is developed for the generalized Schr{ö}dinger equation with random fluctuations. The wave field $ ψ $ is expanded in terms of eigenfunctions: $ ψ = \sum_{n} a_{n} (t) ϕ_{n} (x) $, with $ ϕ_{n} $ being the eigenfunction that satisfies the eigenvalue equation $ H_{0} ϕ_{n} = λ_{n} ϕ_{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.