# On the pointwise convergence of the integral kernels in the Feynman-Trotter formula

## TL;DR

This paper proves that the integral kernels of Trotter-type path integral approximations for Schrödinger equations with certain potentials converge pointwise, except at specific exceptional times, using harmonic analysis and pseudo-differential operator techniques.

## Contribution

It establishes the pointwise convergence of integral kernels in the Feynman-Trotter formula for a broad class of potentials, extending understanding beyond strong operator convergence.

## Key findings

- Pointwise kernel convergence occurs uniformly on compact sets.
- Convergence fails at certain exceptional times where kernels are distributions.
- Results apply to potentials in various harmonic analysis function spaces.

## Abstract

We study path integrals in the Trotter-type form for the Schr\"odinger equation, where the Hamiltonian is the Weyl quantization of a real-valued quadratic form perturbed by a potential $V$ in a class encompassing that - considered by Albeverio and It\^o in celebrated papers - of Fourier transforms of complex measures. Essentially, $V$ is bounded and has the regularity of a function whose Fourier transform is in $L^1$. Whereas the strong convergence in $L^2$ in the Trotter formula, as well as several related issues at the operator norm level are well understood, the original Feynman's idea concerned the subtler and widely open problem of the pointwise convergence of the corresponding probability amplitudes, that are the integral kernels of the approximation operators. We prove that, for the above class of potentials, such a convergence at the level of the integral kernels in fact occurs, uniformly on compact subsets and for every fixed time, except for certain exceptional time values for which the kernels are in general just distributions. Actually, theorems are stated for potentials in several function spaces arising in Harmonic Analysis, with corresponding convergence results. Proofs rely on Banach algebras techniques for pseudo-differential operators acting on such function spaces.