Carnot geometry and the resolvent of the sub-Laplacian for the Heisenberg group

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We obtain an explicit representation formula for the sub-Laplacian on the Isotropic, three-dimensional Heisenberg group. Using the formula we obtain themeromorphic continuation of the resolvent to the logarithmic plane, the existence of boundary values in the continuous spectrum, and semiclassical asymptotics of the resolvent kernel. The asymptotic formulas show the contribution of each Hamiltonian path in Carnot geometry to the spatial and high-energy asymptotics of the resolvent (convolution) kernel for the sub-Laplacian.

Original languageEnglish
Pages (from-to)745-769
Number of pages25
JournalCommunications in Partial Differential Equations
Issue number3-4
StatePublished - 2003

Bibliographical note

Funding Information:
I am grateful to Richard Beals, Ruth Gornet, Scott Pauls, and Jack Schmidt for valuable discussions, and to the Mathematics Department at Dartmouth College for its hospitality during part of the time that this work was done. Supported in part by NSF Grant DMS-0100829.


  • Carnot geometry
  • Resolvent kernel
  • Semiclassical asymptotics
  • Sub-Laplacian

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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