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CONDITIONAL FEYNMAN INTEGRAL AND SCHRÖDINGER INTEGRAL EQUATION ON A FUNCTION SPACE

  • DONG HYUN CHO (a1)
Abstract
Abstract

Let Cr[0,t] be the function space of the vector-valued continuous paths x:[0,t]→ℝr and define Xt:Cr[0,t]→ℝ(n+1)r by Xt(x)=(x(0),x(t1),…,x(tn)), where 0<t1<⋯<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[FtXt] of Ft given by where θ(s,⋅) are the Fourier–Stieltjes transforms of the complex Borel measures on ℝr, and provide an inversion formula for Eanfq[FtXt]. 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[FtXt] and a probability distribution on ℝr when Xt(x)=(x(0),x(t)).

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This work was supported by Kyonggi University Research Grant.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2] R. H. Cameron and D. A. Storvick , ‘Analytic Feynman integral solutions of an integral equation related to the Schrödinger equation’, J. Analyse Math. 38 (1980), 3466.

[4] D. H. Cho , ‘A simple formula for an analogue of conditional Wiener integrals and its applications’, Trans. Amer. Math. Soc. 360(7) (2008), 37953811.

[7] D. M. Chung and D. L. Skoug , ‘Conditional analytic Feynman integrals and a related Schrödinger integral equation’, SIAM J. Math. Anal. 20(4) (1989), 950965.

[8] M. K. Im and K. S. Ryu , ‘An analogue of Wiener measure and its applications’, J. Korean Math. Soc. 39(5) (2002), 801819.

[9] G. W. Johnson and D. L. Skoug , ‘Notes on the Feynman integral, III: The Schroedinger equation’, Pacific J. Math. 105(2) (1983), 321358.

[10] C. Park and D. L. Skoug , ‘A simple formula for conditional Wiener integrals with applications’, Pacific J. Math. 135(2) (1988), 381394.

[11] K. S. Ryu and M. K. Im , ‘A measure-valued analogue of Wiener measure and the measure-valued Feynman–Kac formula’, Trans. Amer. Math. Soc. 354(12) (2002), 49214951.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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