Sequential Fourier-Feynman transform, convolution and first variation

Abstract

Abstract. Cameron and Storvick introduced the concept of a sequential Fourier-Feynman transform and established the existence of this transform for functionals in a Banach algebraŜ of bounded functionals on classical Wiener space. In this paper we investigate various relationships between the sequential Fourier-Feynman transform and the convolution product for functionals which need not be bounded or continuous. Also we study the relationships involving this transform and the first variation.

Integration by parts formulas for analytic Feynman integrals of unbounded functionals

“…is given by Equation (11) and φ ∈M 1 (R n ) is given by Equation (12). Then for any w ∈ A μ , the first variation δF (y|w) exists and is given by the formula…”

“…Because = ψ + φ in Equation (13), there are three cases as in Theorems 2.6-2.8, and finally we put together these three cases in Corollary 2.9. THEOREM 2.6 Let F l (x) = G l (x)ψ l ((e, x) ∼ ), where G l ∈ F(B) is given by Equation (11) with corresponding measure μ l and ψ l ∈ L 1,2 1 (R n ) ∩ L 2 (R n ) for l = 1, 2. Then for…”

“…is given by (11) with corresponding measure μ l for l = 1, 2, ψ ∈ L 1 1 (R n ) and φ ∈M 1 (R n ) with corresponding measure σ . Then for each w ∈ A μ 1 ∩ A μ 2 , the integration by parts formula (34) holds.…”

“…Functionals of this type were *Corresponding author. Email: mathkbs@snut.ac.kr studied in [1][2][3][10][11][12][18][19][20]. In the applications of the Feynman integral to the quantum theory, the function in Equation (1) corresponds to the initial condition associated with the Schrödinger equation.…”

“…on B. It is well known [13,14] that F(B) is a Banach algebra and the mapping μ −→ G is a Banach algebra isomorphism where μ and G are related by Equation (11). LetM(R n ) be the set of functions φ defined on R n by…”