Bulletin of the Australian Mathematical Society

Research Article

CONDITIONAL FEYNMAN INTEGRAL AND SCHRÖDINGER INTEGRAL EQUATION ON A FUNCTION SPACE

DONG HYUN CHOa1

a1 Department of Mathematics, Kyonggi University, Kyonggido Suwon 443-760, Korea (email: j94385@kyonggi.ac.kr)

Abstract

Let Cr[0,t] be the function space of the vector-valued continuous paths x:[0,t]→xs211Dr and define Xt:Cr[0,t]→xs211D(n+1)r by Xt(x)=(x(0),x(t1),…,x(tn)), where 0<t1<xs22EF<tn=t. In this paper, using a simple formula for the conditional expectations of the functions on Cr[0,t] given Xt, we evaluate the conditional analytic Feynman integral Eanfq[Ftxs2223Xt] of Ft given by

\[ F_t(x) = \exp \biggl \{\int _0^t \theta (s, x(s)) \,ds \biggr \} \quad \mbox {for } x \in C^r [0, t], \]

where θ(s,xs22C5) are the Fourier–Stieltjes transforms of the complex Borel measures on xs211Dr, and provide an inversion formula for Eanfq[Ftxs2223Xt]. Then we present an existence theorem for the solution of an integral equation including the integral equation which is formally equivalent to the Schrödinger differential equation. We show that the solution can be expressed by Eanfq[Ftxs2223Xt] and a probability distribution on xs211Dr when Xt(x)=(x(0),x(t)).

(Received March 03 2008)

2000 Mathematics subject classification

  • primary 28C20

Keywords and phrases

  • analytic Feynman integral;
  • conditional analytic Feynman integral;
  • Schrödinger equation;
  • time-dependent potential;
  • Wiener space

Footnotes

This work was supported by Kyonggi University Research Grant.