Abstract
AbstractWe give a construction of the heat kernel and Green function of a hypoelliptic operator on the one-dimensional Heisenberg group \(\mathbb{H}\), the sub-Laplacian \(\mathcal{L}\). The explicit formulas are developed using Fourier–Wigner transforms, pseudo-differential operators of the Weyl type, i.e., Weyl transforms, and spectral analysis. These formulas are obtained by first finding the formulas for the heat kernels and Green functions of a family of twisted Laplacians \({L}_{\tau}\) for all non-zero real numbers \({\tau}\). In the case when \({\tau=1, {L}_{1}}\) is just the usual twisted Laplacian.KeywordsHeisenberg groupsub-Laplaciantwisted LaplaciansHermite functionsWeyl transformsheat kernelsGreen functions.
Published Version
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