Abstract
In this note we present a notion of harmonic oscillator on the Heisenberg group H n which forms the natural analogue of the harmonic oscillator on ℝ n under a few reasonable assumptions: the harmonic oscillator on H n should be a negative sum of squares of operators related to the sub-Laplacian on H n , essentially self-adjoint with purely discrete spectrum, and its eigenvectors should be smooth functions and form an orthonormal basis of L 2 (H n ). This approach leads to a differential operator on H n which is determined by the (stratified) Dynin–Folland Lie algebra. We provide an explicit expression for the operator as well as an asymptotic estimate for its eigenvalues.
Highlights
The aim of this note is to introduce a canonical harmonic oscillator on the Heisenberg group Hn
The Schrödinger representation ρ1 of Hn acting on L2(Rn) and the associated Lie algebra representation, naturally acting on S (Rn), clearly relate each of the realisations (R1)–(R3) to the others
It ought to be natural to assume that similar realisations should be available for the canonical harmonic oscillator on Hn
Summary
It ought to be natural to assume that similar realisations should be available for the canonical harmonic oscillator on Hn. The special role of the Heisenberg Lie algebra hn in this context is not coincidental: it is precisely the Lie algebra which is generated by the partial derivatives ∂tj and the multiplication operators for the coordinate functions tk , j , k = 1, . Endowed with the canonical homogeneous structure arising from the stratification, the Dynin–Folland Lie algebra together with its associated connected, connected Lie group, the group’s generic irreducible unitary representations, and the associated negative sub-Laplacian (a positive Rockland operator) give rise to the harmonic oscillator on the Heisenberg group. 1extended to the universal enveloping algebra u(hn ). 2The factor 4π2 is due to our choice of realising the Schrödinger representation; our expression agrees with the versions in Folland [4] or Stein [12] up to scaling
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