Abstract
We continue our analysis of nilpotent groups related to quantum mechanical systems whose Hamiltonians have polynomial interactions. For the spinless particle in a constant external magnetic field, the associated nilpotent group is the Heisenberg group. We solve the heat equation for the Heisenberg group by diagonalizing the sub-Laplacian. The unitary map to the Hilbert space in which the sub-Laplacian is a multiplication operator with positive spectrum is given. The spectral multiplicity is shown to be related to the irreducible representations of SU(2). A Lax pair, generated from the Heisenberg sub-Laplacian, is used to find operators unitarily equivalent to the sub-Laplacian, but not arising from the SL(2,R) automorphisms of the Heisenberg group.
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Department of Mathematics, supported in part by NSF.
Department of Physics and Astronomy, supported in part by DOE.
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Jorgensen, P.E.T., Klink, W.H. Spectral transform for the sub-Laplacian on the Heisenberg group. J. Anal. Math. 50, 101–121 (1988). https://doi.org/10.1007/BF02796116
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DOI: https://doi.org/10.1007/BF02796116