title: Stochastic approach to the generalized Schrödinger equation: A method of eigenfunction expansion.
abstract: Using a method of eigenfunction expansion, a stochastic equation is developed for the generalized Schrödinger equation with random fluctuations. The wave field ψ is expanded in terms of eigenfunctions: ψ=∑(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 the "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 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.
authors.affiliation: Department of Physics, Ritsumeikan University-BKC, Noji, Kusatsu City, 525-8577 Shiga, Japan.
authors.forename: Satoshi
authors.initials: S
authors.lastname: Tsuchida
authors.affiliation: Department of Physics, Ritsumeikan University-BKC, Noji, Kusatsu City, 525-8577 Shiga, Japan.
authors.forename: Hiroshi
authors.initials: H
authors.lastname: Kuratsuji