THE HARMONIC OSCILLATOR ON THE HEISENBERG GROUP

The Euclidean harmonic oscillator - \Delta + 4 \pi^2 |t|^2 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 \mathbf{H}_n 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 \mathbf{H}_n should be a negative sum of squares of operators related to the Heisenberg sub-Laplacian on \mathbf{H}_n, essentially self-adjoint with purely discrete spectrum, and its eigenvectors should be smooth functions and form an orthonormal basis of L^2(\mathbf{H}_n).
This approach leads to a differential operator on \mathbf{H}_n which is determined by the (stratified) Dynin-Folland Lie algebra (cf. [1, 2, 3, 6]) and its generic unitary representations, which act on L^2(\mathbf{H}_n). In the simplest (3-dimensional) case this operator is given by

\mathcal{Q}_{\mathbf{H}_1} = - \mathcal{L}_{\mathbf{H}_1} + 4 \pi^2 t_3^2
=-\left( \partial_{t_1}^2 + \partial_{t_2}^2 \right) - \frac{1}{4} \left ({t_1}^2 + {t_2}^2 \right ) \partial_{t_3}^2 - \left( t_1 \partial_{t_2} - t_2 \partial_{t_1} \right ) \partial_{t_3} + 4 \pi^2 t_3^2.

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 \mathcal{Q}_{\mathbf{H}_n} on the Heisenberg group \mathbf{H}_n has a purely discrete spectrum \mathrm{spec}(\mathcal{Q}_{\mathbf{H}_n}) \subset (0, \infty). The number of its eigenvalues, counted with multiplicities, which are less or equal to \lambda > 0 is asymptotically given by

N(\lambda) \sim \lambda^{\frac{6n + 3}{2}},

and the magnitude of the eigenvalues is asymptotically equal to

\lambda_{k} \sim k^{ \frac{2}{6n+3}} \text{ for } k = 1, 2, \ldots.

Moreover, the eigenvectors of \mathcal{Q}_{\mathbf{H}_n} are in \mathcal{S}(\mathbf{H}_n) and form an orthonormal basis of L^{2}(\mathbf{H}_n).

The power {\frac{6n + 3}{2}} bears a specific relation to the canonical homogeneous structure of \mathfrak{h}_n: the nominator 6n + 3 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.

References:

  1. A. S. Dynin. Pseudodifferential operators on the Heisenberg group. Dokl. Akad. Nauk SSSR, 225:1245–1248, 1975.
  2. G. B. Folland. Meta-Heisenberg groups. In Fourier analysis (Orono, ME, 1992), volume 157 of Lecture Notes in Pure and Appl. Math., pages 121–147. Dekker, New York, 1994.
  3. V. Fischer, D. Rottensteiner, and M. Ruzhansky. Heisenberg-Modulation Space on the Crossroads of Coorbit Theory and Decomposition Space Theory. Preprint, 2018. https://arxiv.org/abs/ 1812.07876.
  4. B. Helffer and J. Nourrigat. Caracterisation des op ́erateurs hypoelliptiques homog`enes invariants `a gauche sur un groupe de Lie nilpotent gradu ́e. Comm. Partial Differential Equations, 4(8):899– 958, 1979.
  5. A. Mohamed, P. L ́evy-Bruhl, and J. Nourrigat. E ́tude spectrale d’op ́erateurs li ́es `a des repr ́esentations de groupes nilpotents. J. Funct. Anal., 113(1):65–93, 1993.
  6. D. Rottensteiner. Time-Frequency Analysis on the Heisenberg Group. PhD thesis, Imperial Col- lege London, September 2014.
  7. D. Rottensteiner and M. Ruzhansky. Harmonic and Anharmonic Oscillators on the Heisenberg Group. Preprint, 2018.
  8. D. Rottensteiner and M. Ruzhansky. The harmonic oscillator on the Heisenberg group. C. R. Math. Acad. Sci. Paris, 358(5):609–61, 2020.
  9. A. F. M. ter Elst and D. W. Robinson. Spectral estimates for positive Rockland operators. In Algebraic groups and Lie groups, volume 9 of Austral. Math. Soc. Lect. Ser., pages 195–213. Cambridge Univ. Press, Cambridge, 1997.