The Heisenberg oscillator

In this short note, we determine the spectrum of the Heisenberg oscillator which is the operator deﬁned as L +| x | 2 +| y | 2 on the Heisenberg group H 1 = R 2 x , y × R where L stands for the positive sublaplacian.

Let X , Y and T be the three elements of h 1 forming the canonical basis of h 1 ; it satisfies [X, Y ] = T . We identify the elements of h 1 with left invariant vector fields on H 1 and we define the sublaplacian: L = −(X 2 + Y 2 ). Let τ be the representation of H 1 on L 2 (R) such that dτ (X ) = ∂ x , dτ (Y ) = i x and necessarily dτ (T ) = i.
Then τ is the well known unitary irreducible Schrödinger representation of H 1 corresponding to the central character t → e it . Furthermore The spectrum of the quantum harmonic is well known and this last equality allows to describe the spectrum of L.
In this short note, we reverse the line of approach described above to study the following unbounded operator on L 2 (H 1 ): we call this operator the Heisenberg oscillator. Our main result is the determination of its spectrum.
This study could very easily be generalised to the (2n + 1)-dimensional Heisenberg group.
In fact we will study the operator L+λ 2 2 (x 2 +y 2 ) for λ 2 = 0, even if by homogeneity it would suffice to study the case λ 2 = 1.
In the Heisenberg oscillator the central variable of H 1 appears only as derivatives in the expression of the vector fields This motivates our choice to study the Heisenberg oscillator intertwined with the Fourier transform F λ 1 in the central variable of H 1 : Hence the object at the centre of this paper is where λ = (λ 1 , λ 2 ) with λ 2 = 0.
The result of this note gives a complete description of the spectrum of the operator (3) which can also be viewed as a magnetic Schrödinger operator with quadratic potential. Some of the properties of the spectrum of that type of operators are already known by specialists of this domain (see for example [4]) and coincide with our explicit description in the particular case of the operator (3). In the future the result of this note will allow the study of a Mehler type formula for the operator given by (3), of the L p -multipliers problem and of Strichartz estimates for the Heisenberg oscillator L + (x 2 + y 2 ).
This paper is organised as follows. First we construct a six-dimensional nilpotent Lie group N and a representation ρ λ of N such that the image of the canonical sublaplacian L of N through ρ λ is given by (3). In the third section we study more systematically the representations of N via the orbit method and the diagonalisation of the image of L. It allows us in the fourth section to go back to the study of the Heisenberg oscillator. In a last section, we obtain a Mehler type formula for the operator given by (3).

The group N
We consider the unbounded operators on L 2 (H 1 ) given by the left-invariant vector fields X and Y (see (1)) and the multiplications by i x and iy. They generate a sixdimensional real Lie algebra whose canonical basis satisfies the commutator relations with all the other commutators vanishing (beside the ones given by skew-symmetry). Hence n is a well defined two-step nilpotent Lie algebra. It is stratified [3] since we can decompose: generates the Lie algebra n and the subspace is the centre of n.
The connected simply connected nilpotent Lie group associated with n is N identified with v × z ∼ R 6 using exponential coordinates. Hence N is endowed with the group law we have: We identify the elements of n with left invariant vector fields on N . We denote by the canonical sublaplacian of N .

The representation ρ λ
Let λ = (λ 1 , λ 2 ) with λ 2 = 0. We consider the representation dρ λ of the Lie algebra n over L 2 (R 2 ) defined by: (5) Throughout this paper, L 2 (R 2 ) is endowed with its natural Hilbert space structure whose Hermitian product is given by: It is not difficult to compute that dρ λ is the infinitesimal representation of the unitary representation ρ λ of N on L 2 (R 2 ) given by: By (5) the image of the canonical sublaplacian L of N (see (4)) through ρ λ is: In the next section, we will show that ρ λ is equivalent to an irreducible unitary representation π λ and we will diagonalise π λ (L).

The representations of N
In this section, after describing all the unitary irreducible representations of N using the orbit method [1], we obtain a diagonalisation of ρ λ (L).

All the representations of N
We need to describe the orbits of N acting on the dual n * of n by the dual of the adjoint action. Each element of n * will be written as = (ω, λ) where ω and λ are linear forms on v and z respectively, identified with a vector of v and z by the canonical scalar products of these two spaces. It is not difficult to determine representatives of the co-adjoint orbits:

Lemma 3.1 Each co-adjoint orbit of N admits exactly one representative of the form
Sketch of the proof For each z ∈ z, let j z be the endomorphism of v given by: where , . v and , . z denote the canonical scalar products on v and z respectively. In the canonical basis As the nilpotent Lie group N is of step two, we compute easily for = (ω, λ) and and the previous paragraph completes the proof.
It is a routine exercise to compute a representation associated with a linear form and we just give here the end result for the linear forms described in Lemma 3.1.
Let λ 2 = 0, λ 1 = 0, ω ∈ RX 2 ⊕ RY 2 as in (ii) of Lemma 3.1. The representation π λ 1 ,ω of N over L 2 (R) given by: is the irreducible unitary representation associated with the linear form given by (ω, λ). Let λ 2 = λ 1 = 0 and ω ∈ v as in (iii) of Lemma 3.1. The character gives the one-dimensional unitary representation associated with the linear form given by ω. By Kirillov's methods, the representations π λ , π λ 1 ,w and e i ω,· exhaust all the irreducible unitary representations of N , up to unitary equivalence.

The representations π λ and ρ λ
Let us focus on the representations π λ with λ = (λ 1 , λ 2 ), λ 2 = 0. Its infinitesimal representation is given by: We can now go back to the study of the representation ρ λ . Its restriction to the centre gives the character z → e iλ(z) ; so by Kirillov's method, we know that ρ λ is equivalent to one or several copies of π λ , depending whether ρ λ is irreducible. In fact it is not difficult to find a concrete expression for the intertwiner between ρ λ and π λ (see the proposition just below) and this shows in particular that ρ λ is irreducible. λ = (λ 1 , λ 2 ), λ 2 = 0, the representations ρ λ and π λ are unitarily equivalent. More precisely, let T λ = T : L 2 (R 2 ) → L 2 (R 2 ) be the unitary operator given by:
Proof The operator T can be written as are the unitary operators given by: The computations of the infinitesimal action on the canonical basis through ρ (1)

Diagonalisation of dπ λ (L)
By (7) the image of the canonical sublaplacian through π λ is the operator: for which we determine a diagonalisation basis. We need to study the homogeneous polynomial of degree two: where and this boils down to diagonalising the matrix M λ . We obtain: and k λ is the orthogonal 2 × 2-matrix: The change of variable transforms the homogeneous polynomial (8) into μ +,λ u 1 2 + μ −,λ u 2 2 and leaves the ); the operator π λ (L) becomes: Recall that the Hermite functions h m , m ∈ N, defined by: form an orthonormal basis of L 2 (R) which diagonalises the quantum harmonic oscillator: Using the notation above, we obtain: The operator π λ (L) admits the following orthonormal basis of eigenfunctions: where m = (m + , m − ) ∈ N 2 and u = k λ u. The eigenvalue associated with h λ,m is ν λ,m := μ 1/2 Consequently, by Proposition 3.2, we obtain: The operator given by (6), that is, admits {T h λ,m , m ∈ N 2 } as orthonormal basis of eigenfunctions and the eigenvalue associated with T h λ,m is ν λ,m .