Jae Gil Choi

Integration by parts formulas involving generalized Fourier-Feynman transforms on function space

In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form F(x) = f( ,..., ) where denotes the Paley-Wiener-Zygmund stochastic integral ∫ T 0 α(t)dx(t).

GENERALIZED FOURIER-FEYNMAN TRANSFORM AND SEQUENTIAL TRANSFORMS ON FUNCTION SPACE

In this paper werst investigate the existence of the gener- alized Fourier-Feynman transform of the functional F given by F ( x) = ^ (( e1;x) � ;:::; ( en;x) � ) ; where ( e;x) � denotes the Paley-Wiener-Zyg stochastic integral with x in a very general function space Ca;b(0 ;T ) and ^ is the Fourier transform of complex measure on B( R n ) withnite total variation. We then dene two sequential transforms. Finally, we establish that the one is to identify the generalized Fourier-Feynman transform and the another transform acts like an inverse generalized Fourier-Feynman transform.

A MULTIPLE GENERALIZED FOURIER–FEYNMAN TRANSFORM VIA A ROTATION ON WIENER SPACE

In this paper we use a rotation property of Wiener measure to define a very general multiple Fourier–Feynman transform on Wiener space. We then proceed to establish its many algebraic properties as well as to establish several relationships between this generalized multiple transform and the corresponding generalized convolution product.

Parts formulas involving conditional generalized Feynman integrals and conditional generalized Fourier–Feynman transforms on function space

In this paper, we first establish a formula for the conditional generalized Feynman integral of the first variation of a functional F(x) with x in a very general function space C a , b [0, T]. We then use this basic formula to obtain several integration by parts formulas for conditional generalized Feynman integrals and conditional generalized Fourier–Feynman transforms. E-mail: jgchoi@dankook.ac.kr E-mail: sejchang@dankook.ac.kr

A Rotation on Wiener Space with Applications

We first investigate a rotation property of Wiener measure on the product of Wiener spaces. Next, using the concept of the generalized analytic Feynman integral, we define a generalized Fourier-Feynman transform and a generalized convolution product for functionals on Wiener space. We then proceed to establish a fundamental result involving the generalized transform and the generalized convolution product.

GENERALIZED ANALYTIC FEYNMAN INTEGRAL VIA FUNCTION SPACE INTEGRAL OF BOUNDED CYLINDER FUNCTIONALS

In this paper, we use a generalized Brownian motion to dene a generalized analytic Feynman integral. We then obtain some results for the generalized analytic Feynman integral of bounded cylinder functionals of the form F(x) = ^ ((g1;x) � ;:::;(gn;x) � ) dened on a very general function space Ca;b(0;T). We also present a change of scale formula for function space integrals of such cylinder func- tionals.

AN APPROACH TO SOLUTION OF THE SCHR¨ ODINGER EQUATION USING FOURIER-TYPE FUNCTIONALS

Abstract. In this paper, we consider the Fourier-type functionals onWiener space. We then establish the analytic Feynman integrals involv-ing the ⋄-convolutions. Further, we give an approach to solution of theSchro¨dinger equation via Fourier-type functionals. Finally, we use this ap-proach to obtain solutions of the Schro¨dinger equations for harmonic oscil-lator and double-well potential. The Schro¨dinger equations for harmonicoscillator and double-well potential are meaningful subjects in quantummechanics. 1. IntroductionLet C 0 [0,T] denote the one-parameter Wiener space, that is, the space ofcontinuous real-valued functions xon [0,T] with x(0) = 0. In 1948, Feynmanassumed the existence of an integral over a space of paths and used this integralin a formal way in his approach to quantum mechanics [9]. A number ofmathematicians have attempted to give rigorously meaningful definitions ofthe Feynman integral with appropriate existence theorems and have expressedsolutions of the Schr¨odinger equation in terms of their integrals. One of theseapproaches is based on the similarity between the Wiener and the Feynmanintegrals, where procedures are developed to obtain the Feynman integralsfrom the Wiener integrals by an analytic extension from the real axis to theimaginary axis.Consider a differential equation(1.1)∂∂tψ(u,t) =12λ∆ψ(u,t) −V(u)ψ(u,t)with the initial condition ψ(u,0) = ϕ(u), where ∆ is the Laplacian and V is anappropriate potential function. For λ>0, this is the diffusion equation with

Sequential Generalized Transforms on Function Space

We define two sequential transforms on a function space induced by generalized Brownian motion process. We then establish the existence of the sequential transforms for functionals in a Banach algebra of functionals on . We also establish that any one of these transforms acts like an inverse transform of the other transform. Finally, we give some remarks about certain relations between our sequential transforms and other well-known transforms on .

MULTIPLE L p ANALYTIC GENERALIZED FOURIER-FEYNMAN TRANSFORM ON THE BANACH ALGEBRA

In this paper, we use a generalized Brownian motion process to deflne a generalized Feynman integral and a general- ized Fourier-Feynman transform. We also deflne the concepts of the multiple Lp analytic generalized Fourier-Feynman transform and the generalized convolution product of functionals on function space Ca;b(0;T). We then verify the existence of the multiple Lp analytic generalized Fourier-Feynman transform for functionals on function space that belong to a Banach algebra S(L 2 (0;T)). Fi- nally we establish some relationships between the multiple Lp an- alytic generalized Fourier-Feynman transform and the generalized convolution product for functionals in S(L 2 (0;T)).

CONDITIONAL GENERALIZED FOURIER-FEYNMAN TRANSFORM AND CONDITIONAL CONVOLUTION PRODUCT ON A BANACH ALGEBRA

In (10), Chang and Skoug used a generalized Brownian motion process to deflne a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this pa- per we deflne the conditional generalized Fourier-Feynman trans- form and conditional generalized convolution product on function space. We then establish some relationships between the condi- tional generalized Fourier-Feynman transform and conditional gen- eralized convolution product for functionals on function space that belonging to a Banach algebra.