The Euclidean harmonic oscillator is a very well understood and frequently appearing partial differential operator. Motivated by our research activities in PDE theory on Lie groups, we investigated what the natural analogous notion of harmonic oscillator on the Heisenberg group should be. The answer we have found is based on a few reasonable assumptions, whose Euclidean versions are satisfied by the classical harmonic oscillator: the harmonic oscillator on should be a negative sum of squares of operators related to the Heisenberg sub-Laplacian on , essentially self-adjoint with purely discrete spectrum, and its eigenvectors should be smooth functions and form an orthonormal basis of .
This approach leads to a differential operator on which is determined by the (stratified) Dynin-Folland Lie algebra (cf. [1, 2, 3, 6]) and its generic unitary representations, which act on . In the simplest (3-dimensional) case this operator is given by
Due to fundamental results on the spectral theory of Rockland operators [4, 5, 9], we could show the following [7, 8].
Theorem. The harmonic oscillator on the Heisenberg group has a purely discrete spectrum . The number of its eigenvalues, counted with multiplicities, which are less or equal to is asymptotically given by
and the magnitude of the eigenvalues is asymptotically equal to
Moreover, the eigenvectors of are in and form an orthonormal basis of .
The power bears a specific relation to the canonical homogeneous structure of : the nominator is related to the homogeneous dimension of the Dynin-Folland Lie algebra, while the denominator 2 is the homogeneous degree of the Heisenberg sub-Laplacian.
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