Feynman integrals as Hida distributions: the case of non-perturbative potentials

Abstract

Feynman integrands are constructed as Hida distributions. For our approach we first have to construct solutions to a corresponding Schroedinger equation with time-dependent potential. This is done by a generalization of the Doss approach to time-dependent potentials. This involves an expectation w.r.t. a complex scaled Brownian motion. As examples polynomial potentials of degree $4n+2, n\in\mathbb N,$ and singular potentials of the form $\frac{1}{|x|^n}, n\in\mathbb N$ and $\frac{1}{x^n}, n\in\mathbb N,$ are worked out.