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The analytic operator-valued Feynman integral via additive functionals of Brownian motion. (English) Zbl 0843.28006

The paper reviews the work recently devoted to extending the definition of the analytic operator-valued Feynman integral to a wider class of perturbations of the Laplace operator. More precisely, perturbations of the Dirichlet form by generalized signed measures \(\mu= \mu_+- \mu_-\) are considered (instead of potentials), such that the perturbed form defines a selfadjoint bounded below operator \(H\). After taking care to extend this to the complex-\(L^2\) setting, it follows that \(\exp(- tH)\) can be continued analytically from \(t> 0\) to imaginary \(t\). Using recent developments in the theory of Dirichlet forms and Markov processes (of which a very clear and concise presentation is included), one can associate with \(\mu_\pm\) positive continuous additive functionals \(A^{\pm}_t(\omega)\), where \(\omega\) is the Brownian motion. Under suitable conditions of “compatibility” of \(\mu_-\) with \(\mu_+\), one can show that a Feynman-Kac representation: \[ (\exp(- tH)f)(x)= P_x(e^{- A_t^{\mu_+}(\omega)+ A^{\mu_-}_t(\omega)} f(\omega(t))) \] holds for \(t> 0\) and \(f\in L^2\). The existence of the analytic operator valued Feynman integral hence follows.

MSC:

28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
60J25 Continuous-time Markov processes on general state spaces
81S40 Path integrals in quantum mechanics
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References:

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