On the convergence of a novel family of time slicing approximation operators for Feynman path integrals

TL;DR

This paper investigates the convergence properties of a new family of time slicing approximation operators for Schrödinger propagators, providing rigorous convergence rates and kernel convergence results within a time-frequency analysis framework.

Contribution

It introduces a manageable time slicing approximation for Schrödinger propagators involving quadratic Hamiltonians plus rough potential perturbations, with proven convergence in operator norm and kernel pointwise.

Findings

Proves convergence in the operator norm topology in L^2.

Establishes pointwise kernel convergence for non-exceptional times.

Provides explicit convergence rates depending on the time slicing mesh size.

Abstract

In this note we study the properties of a sequence of approximate propagators for the Schr\"odinger equation, in the spirit of Feynman's path integrals. Precisely, we consider Hamiltonian operators arising as the Weyl quantization of a quadratic form in phase space, plus a bounded potential perturbation in the form of a pseudodifferential operator with a rough symbol. It is known that the corresponding Schr\"odinger propagator is a generalized metaplectic operator. This naturally motivates the introduction of a manageable time slicing approximation consisting of operators of the same type. By means of techniques and function spaces of time-frequency analysis it is possible to obtain several convergence results with precise rates in terms of the mesh size of the time slicing subdivision. In particular, we prove convergence in the norm operator topology in $L^2$, as well as pointwise convergence of the corresponding integral kernels for non-exceptional times.