Chernoff solutions of the heat and the Schr\"odinger equation in the Heisenberg group

Abstract

This paper investigates the application of the classical Chernoff's theorem to construct explicit solutions for the heat and Schr\"odinger 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 Schr\"odinger 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.