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

TL;DR

This paper constructs Feynman integrals as Hida distributions by solving time-dependent Schrödinger equations with complex potentials, extending existing methods to include non-perturbative and singular potentials.

Contribution

It generalizes the Doss approach to time-dependent potentials and constructs Feynman integrals as Hida distributions for complex, polynomial, and singular potentials.

Findings

Successfully constructs solutions for polynomial and singular potentials.

Extends the Doss approach to time-dependent potentials.

Provides explicit examples 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.