• Title, Summary, Keyword: generalized analytic Feynman integral

Search Result 29, Processing Time 0.033 seconds

GENERALIZED ANALYTIC FEYNMAN INTEGRALS INVOLVING GENERALIZED ANALYTIC FOURIER-FEYNMAN TRANSFORMS AND GENERALIZED INTEGRAL TRANSFORMS

  • Chang, Seung Jun;Chung, Hyun Soo
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.21 no.2
    • /
    • pp.231-246
    • /
    • 2008
  • In this paper, we use a generalized Brownian motion process to define a generalized analytic Feynman integral. We then establish several integration formulas for generalized analytic Feynman integrals generalized analytic Fourier-Feynman transforms and generalized integral transforms of functionals in the class of functionals ${\mathbb{E}}_0$. Finally, we use these integration formulas to obtain several generalized Feynman integrals involving the generalized analytic Fourier-Feynman transform and the generalized integral transform of functionals in ${\mathbb{E}}_0$.

  • PDF

A FUBINI THEOREM FOR GENERALIZED ANALYTIC FEYNMAN INTEGRALS AND FOURIER-FEYNMAN TRANSFORMS ON FUNCTION SPACE

  • Chang, Seung-Jun;Lee, Il-Yong
    • Bulletin of the Korean Mathematical Society
    • /
    • v.40 no.3
    • /
    • pp.437-456
    • /
    • 2003
  • In this paper we use a generalized Brownian motion process to define a generalized analytic Feynman integral. We then establish a Fubini theorem for the function space integral and generalized analytic Feynman integral of a functional F belonging to Banach algebra $S(L^2_{a,b}[0,T])$ and we proceed to obtain several integration formulas. Finally, we use this Fubini theorem to obtain several Feynman integration formulas involving analytic generalized Fourier-Feynman transforms. These results subsume similar known results obtained by Huffman, Skoug and Storvick for the standard Wiener process.

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

  • Chang, Seung-Jun;Choi, Jae-Gil
    • Bulletin of the Korean Mathematical Society
    • /
    • v.41 no.1
    • /
    • pp.73-93
    • /
    • 2004
  • In [10], 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 define the conditional generalized Fourier-Feynman transform and conditional generalized convolution product on function space. We then establish some relationships between the conditional generalized Fourier-Feynman transform and conditional generalized convolution product for functionals on function space that belonging to a Banach algebra.

MULTIPLE Lp ANALYTIC GENERALIZED FOURIER-FEYNMAN TRANSFORM ON THE BANACH ALGEBRA

  • Chang, Seung-Jun;Choi, Jae-Gil
    • Communications of the Korean Mathematical Society
    • /
    • v.19 no.1
    • /
    • pp.93-111
    • /
    • 2004
  • In this paper, we use a generalized Brownian motion process to define a generalized Feynman integral and a generalized Fourier-Feynman transform. We also define the concepts of the multiple Lp analytic generalized Fourier-Feynman transform and the generalized convolution product of functional on function space $C_{a,\;b}[0,\;T]$. We then verify the existence of the multiple $L_{p}$ analytic generalized Fourier-Feynman transform for functional on function space that belong to a Banach algebra $S({L_{a,\;b}}^{2}[0, T])$. Finally we establish some relationships between the multiple $L_{p}$ analytic generalized Fourier-Feynman transform and the generalized convolution product for functionals in $S({L_{a,\;b}}^{2}[0, T])$.

A FUBINI THEOREM FOR GENERALIZED ANALYTIC FEYNMAN INTEGRAL ON FUNCTION SPACE

  • Lee, Il Yong;Choi, Jae Gil;Chang, Seung Jun
    • Bulletin of the Korean Mathematical Society
    • /
    • v.50 no.1
    • /
    • pp.217-231
    • /
    • 2013
  • In this paper we establish a Fubini theorem for generalized analytic Feynman integral and $L_1$ generalized analytic Fourier-Feynman transform for the functional of the form $$F(x)=f({\langle}{\alpha}_1,\;x{\rangle},\;{\cdots},\;{\langle}{{\alpha}_m,\;x{\rangle}),$$ where {${\alpha}_1$, ${\cdots}$, ${\alpha}_m$} is an orthonormal set of functions from $L_{a,b}^2[0,T]$. We then obtain several generalized analytic Feynman integration formulas involving generalized analytic Fourier-Feynman transforms.

ANALYTIC OPERATOR-VALUED GENERALIZED FEYNMAN INTEGRALS ON FUNCTION SPACE

  • Chang, Seung Jun;Lee, Il Yong
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.23 no.1
    • /
    • pp.37-48
    • /
    • 2010
  • In this paper we use a generalized Brownian motion process to defined an analytic operator-valued generalized Feynman integral. We then obtain explicit formulas for the analytic operatorvalued generalized Feynman integrals for functionals of the form $$F(x)=f\({\int}^T_0{\alpha}_1(t)dx(t),{\cdots},{\int}_0^T{\alpha}_n(t)dx(t)\)$$, where x is a continuous function on [0, T] and {${\alpha}_1,{\cdots},{\alpha}_n$} is an orthonormal set of functions from ($L^2_{a,b}[0,T]$, ${\parallel}{\cdot}{\parallel}_{a,b}$).

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

  • Chang, Seung-Jun;Choi, Jae-Gil;Chung, Hyun-Soo
    • Bulletin of the Korean Mathematical Society
    • /
    • v.48 no.3
    • /
    • pp.475-489
    • /
    • 2011
  • In this paper, we use a generalized Brownian motion to define 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) = $\hat{v}$(($g_1,x)^{\sim}$,..., $(g_n,x)^{\sim}$) defined on a very general function space $C_{a,b}$[0,T]. We also present a change of scale formula for function space integrals of such cylinder functionals.

Convolution product and generalized analytic Fourier-Feynman transforms

  • Chang, Seung-Jun
    • Communications of the Korean Mathematical Society
    • /
    • v.11 no.3
    • /
    • pp.707-723
    • /
    • 1996
  • We first define the concept of the generalized analytic Fourier-Feynman transforms of a class of functionals on function space induced by a generalized Brownian motion process and study of functionals which plays on important role in physical problem of the form $ F(x) = {\int^{T}_{0} f(t, x(t))dt} $ where f is a complex-valued function on $[0, T] \times R$. We next show that the generalized analytic Fourier-Feynman transform of the convolution product is a product of generalized analytic Fourier-Feynman transform of functionals on functin space.

  • PDF

A TRANSLATION THEOREM FOR THE GENERALIZED FOURIER-FEYNMAN TRANSFORM ASSOCIATED WITH GAUSSIAN PROCESS ON FUNCTION SPACE

  • Chang, Seung Jun;Choi, Jae Gil;Ko, Ae Young
    • Journal of the Korean Mathematical Society
    • /
    • v.53 no.5
    • /
    • pp.991-1017
    • /
    • 2016
  • In this paper we define a generalized analytic Fourier-Feynman transform associated with Gaussian process on the function space $C_{a,b}[0,T]$. We establish the existence of the generalized analytic Fourier-Feynman transform for certain bounded functionals on $C_{a,b}[0,T]$. We then proceed to establish a translation theorem for the generalized transform associated with Gaussian process.

OPERATOR-VALUED FUNCTION SPACE INTEGRALS VIA CONDITIONAL INTEGRALS ON AN ANALOGUE WIENER SPACE II

  • Cho, Dong Hyun
    • Bulletin of the Korean Mathematical Society
    • /
    • v.53 no.3
    • /
    • pp.903-924
    • /
    • 2016
  • In the present paper, using a simple formula for the conditional expectations given a generalized conditioning function over an analogue of vector-valued Wiener space, we prove that the analytic operator-valued Feynman integrals of certain classes of functions over the space can be expressed by the conditional analytic Feynman integrals of the functions. We then provide the conditional analytic Feynman integrals of several functions which are the kernels of the analytic operator-valued Feynman integrals.