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

100%

Cho D.

Open Mathematics

2010

tom 8

nr 5

908-927

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

51%

Kim B., Yoo I., Cho D.

Open Mathematics

2010

tom 8

nr 3

616-632

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.

Ograniczanie wyników

2 Open Mathematics

2 Cho D.

1 Kim B.

1 Yoo I.

2 2010

Wyniki wyszukiwania

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

Fourier-Feynman transforms of unbounded functionals on abstract Wiener space