From Feynman-Kac formula to Feynman integrals via analytic continuation. (English) Zbl 0812.60052

Summary: By using a calculus based on Brownian bridge measures, it is shown that under mild assumptions on \(V\) (e.g. \(V\) is in the Kato class) the fundamental solution (FS) \(q(t,x,y)\) for the heat equation \(\partial_ tu = ({1 \over 2} \Delta - V)u\) can be represented by the Feynman-Kac formula. Furthermore, it has an analytic continuation in \(t\) over \(\mathbb{C}_ +\), where \(\mathbb{C}_ + = \{z \in \mathbb{C}\), \(\text{Re} z>0\}\), and \(q(\varepsilon + it\), \(x,y)\) can be expressed via Wiener path integrals. For small \(\varepsilon>0\) it can be considered as an approximation of the FS for the Schrödinger equation \(\partial_ t \psi = i ({1 \over 2} \Delta - V) \psi\). We also give an estimate of \(q(t,x,y)\) for \(t \in \mathbb{C}_ +\).


60H30 Applications of stochastic analysis (to PDEs, etc.)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
60J65 Brownian motion
60J35 Transition functions, generators and resolvents
81S40 Path integrals in quantum mechanics
Full Text: DOI


[1] Albeverio, S.; Blanchard, Ph.; Ma, Z. M., Singular Schrödinger operators associated with Feynman-Kac semigroups, (Proc. Random Partial Differential Equations, International Series of Numerical Math., Vol. 102 (1991), Birkhäuser)
[2] Albeverio, S.; Høegh-Krohn, R., Mathematical Theory of Feynman Path Integrals, (Springer Lecture Notes in Mathematics, Vol. 523 (1976), Springer: Springer Berlin) · Zbl 0337.28009
[3] Albeverio, S.; Johnson, G. W.; Ma, Z. M., The analytic operator-valued Feynman-integral via additive functionals of Brownian motion, BiBos-Preprint (1992) · Zbl 0843.28006
[4] Blanchard, Ph.; Ma, Z. M., Semigroup of Schrödinger operators with potentials given by Radon measures, (Albeverio, S., Stochastic Processes, Physics and Geometry (1990), World Scientific: World Scientific Singapore), 160-195
[5] Blumenthal, R. M.; Getoor, R. K., Markov Processes and Potential Theory (1968), Academic Press: Academic Press New York · Zbl 0169.49204
[6] Dynkin, E. B., Markov Process I (1965), Springer: Springer Berlin · Zbl 0132.37901
[7] Exner, P., Open Quantum Systems and Feynman Integrals (1985), Reidel Dorderecht · Zbl 0638.46051
[8] Feynman, R. P., Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys., 20, 367-387 (1948) · Zbl 1371.81126
[9] Fukushima, M., Dirichlet Forms and Markov Process (1980), North-Holland and Kodansha
[10] Johnson, W. G., Existence theorems for the analytic operator-valued Feynman integral, Monograph, No. 20 (1988), Univ. of Sherbrooke Series · Zbl 0693.46050
[11] Kac, M., On some connections between probability theory and differential equations, Proc. 2nd Berk. Symp. Math. Stat. and Prob., 189-215 (1950)
[12] Kallianpur, G.; Kannan, D.; Karandikar, R. L., Analytic and sequential Feynman integrals on abstract Wiener and Hilbert spaces, and a Cameron-Martin formula, Ann. Instit. Henri Poincaré, 21, 323-361 (1985) · Zbl 0583.60049
[13] Simon, B., Schrödinger semigroups, Bull Amer. Math. Soc., 7, 447-526 (1982) · Zbl 0524.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.