Chernoff solutions of the heat and the Schrödinger equation in the Heisenberg group
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909550112669696 |
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| author | Drago, Nicolò Mazzucchi, Sonia Pinamonti, Andrea |
| author_facet | Drago, Nicolò Mazzucchi, Sonia Pinamonti, Andrea |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18481 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chernoff solutions of the heat and the Schrödinger equation in the Heisenberg group Drago, Nicolò Mazzucchi, Sonia Pinamonti, Andrea Analysis of PDEs Probability 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. |
| title | Chernoff solutions of the heat and the Schrödinger equation in the Heisenberg group |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2503.18481 |