Wiener path integrals and the fundamental solution for the Heisenberg Laplacian
type: Item
instance of: scholarly article
title: Wiener path integrals and the fundamental solution for the Heisenberg Laplacian
zbMATH Open document ID: 1063.35062
DOI: 10.1007/BF02788796
author: Wolfgang Staubach
published in: Journal d'Analyse Mathématique
publication date: 2004-09-27T00:00:00Z
review text: The author presents explicit calculations of the heat kernel, fundamental solution and Schwartz kernel of the resolvent for the Heisenberg Laplacian \[ \Delta_H= (X_1)^2+ (X_2)^2\;(X_1=\partial/\partial x_1- x_2\partial/\partial x_0,\;X_2= \partial/\partial x_2- x_1\partial/\partial x_0). \] He starts with the Fourier transform of the heat equation \(v_t= \Delta_H v/2\) which provides the heat semigroup \(\exp(-(- \Delta+ \xi^2|c\dot|^2)\) with a quadratic potential. Then the Feynman-Kac formula, the Trotter product formula for self-adjoint operators and Wiener path integrals are applied to obtain the Mehler's formula for the transform \(\widehat v\). Finally, \(u= -\int^\infty_0 v\,dt\) yields the heat kernel.
Mathematics Subject Classification ID: 35H20
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 2103503
zbMATH Keywords: Heisenberg Laplacian
zbMATH Keywords: Wiener path integral
zbMATH Keywords: Heisenberg heat equation
reviewed by: Jan Chrastina
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2103503