On the time slicing approximation of Feynman path integrals for non-smooth potentials

TL;DR

This paper extends the convergence results of time slicing approximations of Feynman path integrals to potentials with limited smoothness, specifically those with second derivatives in a Sobolev space, using a non-perturbative approach.

Contribution

It proves that convergence in the operator norm persists for less smooth potentials, broadening the applicability of Feynman path integral approximations.

Findings

Convergence in operator norm for potentials with Sobolev regularity

Non-perturbative proof techniques used

Applicable to potentials with second derivatives in H^{d+1}

Abstract

We consider the time slicing approximations of Feynman path integrals, constructed via piecewice classical paths. A detailed study of the convergence in the norm operator topology, in the space $\mathcal{B}(L^2(\mathbb{R}^d))$ of bounded operators on $L^2$, and even in finer operator topologies, was carried on by D. Fujiwara in the case of smooth potentials with an at most quadratic growth. In the present paper we show that the result about the convergence in $\mathcal{B}(L^2(\mathbb{R}^d))$ remains valid if the potential is only assumed to have second space derivatives in the Sobolev space $H^{d+1}(\mathbb{R}^d)$ (locally and uniformly), uniformly in time. The proof is non-perturbative in nature, but relies on a precise short time analysis of the Hamiltonian flow at this Sobolev regularity and on the continuity in $L^2$ of certain oscillatory integral operators with non-smooth phase and amplitude.