On the Convergence of a Novel Time-Slicing Approximation Scheme for Feynman Path Integrals.

The article focuses on the development and analysis of a novel time-slicing approximation scheme for the Schrödinger propagator associated with Hamiltonians given by the Weyl quantization of quadratic forms plus bounded potential perturbations with rough symbols in the Sjöstrand class \( M^{\infty,1}(\mathbb{R}^{2d}) \). Building on the framework of generalized metaplectic operators (the class \( FIO(S) \)), the authors introduce approximate propagators whose symbols better mimic the Dyson–Phillips expansion of the exact propagator, achieving a second-order short-time approximation error in the modulation space norm. This leads to convergence results with explicit rates in the uniform operator topology on modulation spaces \( M^p(\mathbb{R}^d) \) for \( 1 \le p \le \infty \), improving upon the classical Trotter formula which only guarantees strong operator convergence without rate control. Furthermore, the paper establishes quantitative convergence of the integral kernels of these approximations to the exact Schrödinger kernel away from exceptional times where the classical flow degenerates, with uniform convergence on compact subsets and rates expressed in terms of the mesh size of the time subdivision. The approach leverages Gabor wave packet analysis and the algebraic properties of the Sjöstrand class, situating the results between the Trotter formula and Fujiwara's oscillatory integral parametrices, and providing a new perspective on operator-theoretic path integral approximations in quantum mechanics.