Chernoff solutions of the heat and the Schrödinger equation in the Heisenberg group

2025/03/24 by Nicolò Drago, Drago, Nicolò, Sonia Mazzucchi +3
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Noncommutative and Quantum Gravity Theories #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2503.18481

openalex publication_date 2025/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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 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.

Related

• Analogues of theorems of Chernoff and Ingham on the Heisenberg group
• Nonlocal heat equations in the Heisenberg group
• Uniqueness of solutions to the Schrodinger equation on the Heisenberg group
• Global representations of the Heat and Schrödinger equation with singular potential
• On theorems of Chernoff and Ingham on the Heisenberg group