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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 \(t \in \mathbb{C}_ +\).

MSC:

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
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References:

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