Abstract
This paper deals with the analytic Feynman integral of functionals on a Wiener space. First the authors establish the existence of the analytic Feynman integrals of functionals in a Banach algebra S α . The authors then obtain a formula for the first variation of integrals. Finally, various analytic Feynman integration formulas involving the first variation are established.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cameron, R. H. and Storvick, D. A., Feynman integral of variation of functionals, Gaussian Random Fields, World Scientific, Singapore, 1980, 144–157.
Cameron, R. H. and Storvick, D. A., Some Banach algebras of analytic Feynman integrable functionals, Analytic Functions, Kozubnik, 1979, Lecture Notes in Math., 798, Springer-Verlag, Berlin, 1980, 18–67.
Cameron, R. H. and Storvick, D. A., Analytic Feynman integral solutions of an integral equation related to the Schrödinger equation, J. Anal. Math., 38, 1980, 34–66.
Cameron, R. H. and Storvick, D. A., Relationships between the Wiener integral and the analytic Feynman integral, Rend. Circ. Mat. Palermo (2) Suppl., 17, 1987, 117–133.
Chang, S. J., Choi, J. G. and Chung, H. S., The approach to solution of the Schrödinger equation using Fourier-type functionals, J. Korean Math. Soc., 50, 2013, 259–274.
Chang, S. J., Chung, H. S. and Skoug, D., Convolution products, integral transforms and inverse integral transforms of functionals in L 2(C0[0, T]), Integral Transforms Spec. Funct., 21, 2010, 143–151.
Chang, K. S., Johnson, G. W. and Skoug, D. L., The Feynman integral of quadratic potentials depending on two time variables, Pacific J. Math., 122, 1986, 11–33.
Chung, H. S. and Chang, S. J., Some applications of the spectral theory for the integral transform involving the spectral representation, J. Funct. Space Appl., 2012, DOI: 10.1155/2012/573602.
Chung, H. S., Choi, J. G. and Chang, S. J., Conditional integral transforms with related topics, Filomat, 26, 2012, 1147–1158.
Chung, H. S., Skoug, D. and Chang, S. J., A Fubini theorem for integral transforms and convolution products, Int. J. Math., 2013, DOI: 10.1142/S0129167X13500249.
Chung, H. S. and Tuan, V. K., Generalized integral transforms and convolution products on function space, Integral Transforms Spec. Funct., 22, 2011, 573–586.
Chung, H. S. and Tuan, V. K., Fourier-type functionals on Wiener space, Bull. Korean Math. Soc., 49, 2012, 609–619.
Chung, H. S. and Tuan, V. K., A sequential analytic Feynman integral of functionals in L 2(C0[0, T]), Integral Transforms Spec. Funct., 23, 2012, 495–502.
Chung, H. S. and Tuan, V. K., Currigendum: Generalized integral transforms and convolution products on function space, Integral Transforms and Special Functions, 24, 2013, 509–509.
Feynman, R. P., Space-time approach to non-relativistic quantum mechanics, Rev. Modern Phys., 20, 1948, 115–142.
Johnson, G. W. and Skoug, D. L., Scale-invariant measurability in Wiener space, Pacific J. Math., 83, 1979, 157–176.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the research fund of Dankook University in 2015.
Rights and permissions
About this article
Cite this article
Chung, H.S., Tuan, V.K. & Chang, S.J. Analytic Feynman integrals of functionals in a Banach algebra involving the first variation. Chin. Ann. Math. Ser. B 37, 281–290 (2016). https://doi.org/10.1007/s11401-016-0967-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-016-0967-3