Chernoff solutions of the heat and the Schrödinger equation in the Heisenberg group
Nicolò Drago, Sonia Mazzucchi, Andrea Pinamonti
TL;DR
This work develops Chernoff-based semigroup approximations for the heat and Schrödinger equations on the Heisenberg group $\mathbb{H}^d$, providing explicit, analytically tractable representations of the fundamental solutions. It constructs Chernoff functions for both the (sub-)Laplacian-driven heat flow and the unitary Schrödinger group, yielding convergence to $e^{tL}$ and $e^{-iHt}$, respectively, and giving rigorous oscillatory-integral and path-integral interpretations in a sub-Riemannian context. A key contribution is the rigorous realization of Brownian motion on $\mathbb{H}^d$ as a weak limit of Chernoff-based random walks, linking probabilistic and analytic pictures of diffusion on nilpotent Lie groups. The paper also clarifies magnetic (vector potential) effects via the magnetic Laplacian, presenting Feynman–Kac–Ito formulas and infinite-dimensional Fresnel integrals that underpin Feynman path integral formulations in this noncommutative setting. Overall, the results advance constructive PDE methods in sub-Riemannian geometry and pave the way for extensions to other nilpotent groups and their associated quantum and stochastic problems.
Abstract
This paper investigates the application of the classical Chernoff's theorem to construct explicit solutions for the heat and Schrödinger equations on the Heisenberg group $\mathbb{H}^d$. Using semigroup approximation techniques, we obtain analytically tractable and numerically implementable representations of fundamental solutions. In particular, we establish a new connection between the heat equation and Brownian motion on $\mathbb{H}^d$ and provide a rigorous realization of the Feynman path integral for the Schrödinger equation. The study highlights the challenges posed by the noncommutative structure of the Heisenberg group and opens new directions for PDEs on sub-Riemannian manifolds.