Analytic Feynman Integral Research Articles

Series expansions of the transform with respect to the Gaussian process

In this paper, we consider the Fourier-type functionals introduced in [Chung HS, Tuan VK. Fourier-type functionals on Wiener space. Bull Korean Math Soc. 2012;49:609–619] and used in [Lee IY, Chung HS, Chang SJ. Series expansions of the analytic Feynman integral for the Fourier-type functional. J Korean Soc Math Educ Ser B. 2012;19:87–102]. We first establish the existence of the transform with respect to the Gaussian process for the Fourier-type functionals. We then obtain the various relationships for the transform with respect to the Gaussian process of the Fourier-type functionals which involved the ⋄-product and in addition, establish various integration formulas.

RELATIONSHIP BETWEEN THE WIENER INTEGRAL AND THE ANALYTIC FEYNMAN INTEGRAL OF CYLINDER FUNCTION

Cameron and Storvick discovered a change of scale for- mula for Wiener integral of functionals in a Banach algebra S on classical Wiener space. We express the analytic Feynman integral of cylinder function as a limit of Wiener integrals. Moreover we obtain the same change of scale formula as Cameron and Storvick's result for Wiener integral of cylinder function. Our result cover a restricted version of the change of scale formula by Kim.

AN APPROACH TO SOLUTION OF THE SCHRÖDINGER EQUATION USING FOURIER-TYPE FUNCTIONALS

In this paper, we consider the Fourier-type functionals on Wiener space. We then establish the analytic Feynman integrals involving the <TEX>${\diamond}$</TEX>-convolutions. Further, we give an approach to solution of the Schr<TEX>$\ddot{o}$</TEX>dinger equation via Fourier-type functionals. Finally, we use this approach to obtain solutions of the Schr<TEX>$\ddot{o}$</TEX>dinger equations for harmonic oscillator and double-well potential. The Schr<TEX>$\ddot{o}$</TEX>dinger equations for harmonic oscillator and double-well potential are meaningful subjects in quantum mechanics.

A FUBINI THEOREM FOR GENERALIZED ANALYTIC FEYNMAN INTEGRAL ON FUNCTION SPACE

In this paper we establish a Fubini theorem for generalized analytic Feynman integral and L1 generalized analytic Fourier-Feynman transform for the functional of the form F(x) = f(h�1,xi,...,hm,xi), where f�1,...,�mg is an orthonormal set of functions from L2 (0,T). We then obtain several generalized analytic Feynman integration formulas involving generalized analytic Fourier-Feynman transforms.

A Cameron-Storvick Theorem for Analytic Feynman Integrals on Product Abstract Wiener Space and Applications

In this paper we derive a Cameron-Storvick theorem for the analytic Feynman integral of functionals on product abstract Wiener space B 2. We then apply our result to obtain an evaluation formula for the analytic Feynman integral of unbounded functionals on B 2. We also present meaningful examples involving functionals which arise naturally in quantum mechanics.

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.

A sequential analytic Feynman integral of functionals in L 2(C 0[0, T

In this paper, we define a sequential analytic Feynman integral of functionals on Wiener space. We then give the existence of the sequential analytic Feynman integral of functionals in L 2(C 0[0, T]). The expansion of functionals in L 2(C 0[0, T]) in terms of Fourier–Hermite functionals plays a key role in the study.

SERIES EXPANSIONS OF THE ANALYTIC FEYNMAN INTEGRAL FOR THE FOURIER-TYPE FUNCTIONAL

In this paper, we consider the Fourier-type functionals introduced in [16]. We then establish the analytic Feynman integral for the Fourier-type functionals. Further, we get a series expansion of the analytic Feynman integral for the Fourier-type functional . We conclude by applying our series expansion to several interesting functionals.

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.

Operator-valued Feynman integral via conditional Feynman integrals on a function space

Abstract Let C 0r [0; t] denote the analogue of the r-dimensional Wiener space, define X t: C r[0; t] → ℝ2r by X t (x) = (x(0); x(t)). In this paper, we introduce a simple formula for the conditional expectations with the conditioning function X t. Using this formula, we evaluate the conditional analytic Feynman integral for the functional $$ \Gamma _t \left( x \right) = exp \left\{ {\int_0^t {\theta \left( {s,x\left( s \right)} \right)d\eta \left( s \right)} } \right\}\varphi \left( {x\left( t \right)} \right) x \in C^r \left[ {0,t} \right] $$, where η is a complex Borel measure on [0, t], and θ(s, ·) and φ are the Fourier-Stieltjes transforms of the complex Borel measures on ℝr. We then introduce an integral transform as an analytic operator-valued Feynman integral over C r [0, t], and evaluate the integral transform for the function Γt via the conditional analytic Feynman integral as a kernel.

Fourier-Feynman transforms of unbounded functionals on abstract Wiener space

Abstract Huffman, Park and Skoug established several results involving Fourier-Feynman transform and convolution for functionals in a Banach algebra S on the classical Wiener space. Chang, Kim and Yoo extended these results to abstract Wiener space for a more generalized Fresnel class $$ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } $$ A1,A2 than the Fresnel class $$ \mathcal{F} $$(B)which corresponds to the Banach algebra S. In this paper we study Fourier-Feynman transform, convolution and first variation of unbounded functionals on abstract Wiener space having the form $$ F\left( x \right) = G\left( x \right)\psi \left( {\left( {\vec e,x} \right)^ \sim } \right) $$, where G∈$$ \mathcal{F} $$(B)and Ψ = ψ + ϕ with ψ ∈ L 1(ℝn) and ϕ is the Fourier transform of a complex Borel measure of bounded variation on ℝn. We also prove a translation theorem for the analytic Feynman integral of the above functionals.

Integration by parts formulas for analytic Feynman integrals of unbounded functionals

Chang, Choi and Skoug established several integration by parts formulas involving generalized Feynman integrals, generalized Fourier–Feynman transforms, and the first variation of cylinder-type functionals. We establish integration by parts formulas for analytic Feynman integrals of unbounded functionals on abstract Wiener space having the form where G belongs to the Fresnel class ℱ(B) and Ψ=ψ+φ with ψ∈L 1(ℝ n ) and φ is the Fourier transform of a complex Borel measure of bounded variation on ℝ n . We also study integration by parts formulas for the analytic Feynman integral involving the Fourier–Feynman transform of those functionals in ℱ(B).

A CHANGE OF SCALE FORMULA FOR CONDITIONAL WIENER INTEGRALS ON CLASSICAL WIENER SPACE

Let <TEX>$X_k(x)=({\int}^T_o{\alpha}_1(s)dx(s),...,{\int}^T_o{\alpha}_k(s)dx(s))\;and\;X_{\tau}(x)=(x(t_1),...,x(t_k))$</TEX> on the classical Wiener space, where <TEX>${{\alpha}_1,...,{\alpha}_k}$</TEX> is an orthonormal subset of <TEX>$L_2$</TEX> [0, T] and <TEX>${\tau}:0<t_1<...<t_k=T$</TEX> is a partition of [0, T]. In this paper, we establish a change of scale formula for conditional Wiener integrals <TEX>$E[G_{\gamma}|X_k]$</TEX> of functions on classical Wiener space having the form <TEX>$$G_{\gamma}(x)=F(x){\Psi}({\int}^T_ov_1(s)dx(s),...,{\int}^T_o\;v_{\gamma}(s)dx(s))$$</TEX>, for <TEX>$F{\in}S\;and\;{\Psi}={\psi}+{\phi}({\psi}{\in}L_p(\mathbb{R}^{\gamma}),\;{\phi}{\in}\hat{M}(\mathbb{R}^{\gamma}))$</TEX>, which need not be bounded or continuous. Here S is a Banach algebra on classical Wiener space and <TEX>$\hat{M}(\mathbb{R}^{\gamma})$</TEX> is the space of Fourier transforms of measures of bounded variation over <TEX>$\mathbb{R}^{\gamma}$</TEX>. As results of the formula, we derive a change of scale formula for the conditional Wiener integrals <TEX>$E[G_{\gamma}|X_{\tau}]\;and\;E[F|X_{\tau}]$</TEX>. Finally, we show that the analytic Feynman integral of F can be expressed as a limit of a change of scale transformation of the conditional Wiener integral of F using an inversion formula which changes the conditional Wiener integral of F to an ordinary Wiener integral of F, and then we obtain another type of change of scale formula for Wiener integrals of F.

Fourier-Feynman transform and first variation of cylinder type functions over Wiener paths in abstract Wiener space

In this paper, we express the analytic Feynman integral of the first variation of cylinder type functions F in terms of the analytic Feynman integral of the product of F with a linear factor over Wiener paths in abstract Wiener space. And then, we derive the Fourier-Feynman transform for the product of cylinder type functions which define the functions in a Banach algebra introduced by Yoo, with n linear factors.

CHANGE OF SCALE FORMULAS FOR WIENER INTEGRAL OVER PATHS IN ABSTRACT WIENER SPACE

Wiener measure and Wiener measurability behave badly under the change of scale transformation. We express the analytic Feynman integral over <TEX>$C_0(B)$</TEX> as a limit of Wiener integrals over <TEX>$C_0(B)$</TEX> and establish change of scale formulas for Wiener integrals over <TEX>$C_0(B)$</TEX> for some functionals.

Analytic Feynman integrals of transforms of variation of cylinder type functions over Wiener paths in abstract Wiener space

Abstract In this paper, we evaluate various analytic Feynman integrals of first variation, conditional first variation, Fourier-Feynman transform and conditional Fourier-Feynman transform of cylinder type functions defined over Wiener paths in abstract Wiener space. We also derive the analytic Feynman integral of the conditional Fourier-Feynman transform for the product of the cylinder type functions which define the functions in a Banach algebra introduced by Yoo, with n linear factors.

Analytic Feynman integrals, Fourier–Feynman transforms and change of scale formula for Wiener integrals over paths on abstract Wiener spaces

We establish the analytic Wiener and Feynman integral theory for Wiener integrals over paths on C 0 (B), the space of abstract Wiener space valued continuous functions on [0, T], for certain cylinder-type functions of the form: where f ∈ L p (R mn ), 1 ≤ p ≤ ∞, and 0 = s 0 ≤ s 1 ≤ · · · ≤ s n = T is a partition of [0, T]. In addition, we establish some relationships between analytic Feynman integrals, Fourier–Feynman transforms and Wiener integrals and we prove the change of scale formula for Wiener integrals over paths on C 0 (B) in abstract Wiener spaces.

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 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.

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

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 <TEX>$S(L^2_{a,b}[0,T])$</TEX> 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.

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 ( ⟨ α 1 , x ⟩ , … , ⟨ α n , x ⟩ ) F(x)=f(\langle \alpha _{1} , x\rangle , \dots , \langle \alpha _{n} , x\rangle ) where ⟨ α , x ⟩ \langle {\alpha ,x}\rangle denotes the Paley-Wiener-Zygmund stochastic integral ∫ 0 T α ( t ) d x ( t ) \int _{0}^{T} \alpha (t) d x(t) .