Green’s Functions For Generalized Schroedinger Equations

Abstract

I. Introduction. The purpose of this paper is to discuss functions defined on the continuous sample paths of Gaussian Markov processes which serve as Green’s functions for pairs of generalized Schroedinger equations. The results extend the author’s earlier paper [2] to a forward time version, and consider different boundary conditions.

A linear transformation theorem for analytic Feynman integrals

“…From (4.9) we see that if μ R is the Gaussian measure on C[a, b] as defined in Theorem 4 then the Radon-Nikodym derivative of μ R with respect to Wiener measure, μ wy is (4.15) for -oo < ξ < oo, We will show that given certain rather strong conditions on ψ, B, and σ the analytic Feynman integral of a certain functional will yield a solution of (4.17). This result differs from those in [3] or [4] in that (4.17) is a generalized Schroedinger equation and from those in [1] in that the solution is expressed in terms of an analytic Feynman integral involving the "derivative" given in (4.16) rather than a generalized analytic Feynman integral. THEOREM [α, b] …”

Sequential Gaussian Markov Integrals

“…Proof Equation (3.1) is true for λ > 0 by (4.1) of [5]. The following integral equation (4.2) is formally analogous to the differential equations (3.3) and (3.4) of [5] for complex λ 9 Reλ> 0.…”

“…The following integral equation (4.2) is formally analogous to the differential equations (3.3) and (3.4) of [5] for complex λ 9 Reλ> 0. Let us proceed in a formal fashion to show that.…”

Gaussian Markov expectations and related integral equations

“…For an appropriate functional, such an integral solves an integral equation related to the Schroedinger equation. The purpose of this paper is to define such integrals for Gaussian Markov stochastic processes, and prove that for appropriate functionals they satisfy an integral equation related to the generalized Schroedinger equation discussed by the first author in [5], [6], [7], and [8]. Examples of Gaussian Markov processes are the Wiener, Ornstein-Uhlenbeck, and Doob-Kac processes.…”

“…where F is a real or complex valued functional defined for all continuous functions on [s, t] and ψ is a real or complex valued function defined almost everywhere on (-co, co) and £ is a real number; of growth conditions are placed on θ and ψ, the expression (2.5) (which now depends on s and ξ) is a solution of a partial differential equation (see [5]). In this case ψ is not necessarily in L 2 .…”