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We study path integrals in the Trotter-type form for the Schr\"odingerequation, where the Hamiltonian is the Weyl quantization of a real-valuedquadratic form perturbed by a potential $V$ in a class encompassing that -considered by Albeverio and It\^o in celebrated papers - of Fourier transformsof complex measures. Essentially, $V$ is bounded and has the regularity of afunction 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 operatornorm level are well understood, the original Feynman's idea concerned thesubtler and widely open problem of the pointwise convergence of thecorresponding probability amplitudes, that are the integral kernels of theapproximation operators. We prove that, for the above class of potentials, sucha convergence at the level of the integral kernels in fact occurs, uniformly oncompact subsets and for every fixed time, except for certain exceptional timevalues for which the kernels are in general just distributions. Actually,theorems are stated for potentials in several function spaces arising inHarmonic Analysis, with corresponding convergence results. Proofs rely onBanach algebras techniques for pseudo-differential operators acting on suchfunction spaces.

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