Wiener path integrals and the fundamental solution for the Heisenberg Laplacian
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.
scientific article; zbMATH DE number 2103503
scientific article