# Comptes Rendus
Mathématique

David Rottensteiner and Michael Ruzhansky

The Harmonic Oscillator on the Heisenberg Group

Volume 358, issue 5 (2020), p. 609-614

Published online: 14 September 2020

https://doi.org/10.5802/crmath.78

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# 1. Introduction

The aim of this note is to introduce a canonical harmonic oscillator on the Heisenberg group $H_n$. For $n = 1$ and exponential coordinates $(t_1, t_2, t_3) \in \mathbb{R}^3 \cong H_1$ the harmonic oscillator we propose is explicitly given by

$$ \mathcal{D}_{H_1} = -(\partial_{t_1}^2 + \partial_{t_2}^2) - \frac{1}{4}(t_1^2 + t_2^2)\partial_{t_3}^2 - (t_1\partial_{t_2} - t_2\partial_{t_1})\partial_{t_3} + 4\pi^2 t_3^2. $$

Our approach is motivated by the following three realisations of the classical harmonic oscillator $\mathcal{D}_{\mathbb{R}^n}$ on $\mathbb{R}^n$:

(R1) the negative sum of squares $-\Delta + 4\pi^2 |t|^2$ of partial derivatives of order 1 and coordinate multiplication operators;

(R2) the Weyl and Kohn-Nirenberg quantizations on $\mathbb{R}^n$ of the symbol $\sigma(t, \xi) := 4\pi^2(|\xi|^2 + |t|^2)$ with $t, \xi \in \mathbb{R}^n$;

(R3) the image $d\rho_1(-\mathcal{L}_{H_n})$ of the negative sub-Laplacian $-\mathcal{L}_{H_n}$ on $H_n$ under the infinitesimal Schrödinger representation $d\rho_1$ (of Planck's constant equal to 1) of the Heisenberg Lie algebra $\mathfrak{h}_n^1$, for $n = 1$.

The operator $\mathcal{D}_{\mathbb{R}^n}$ is usually defined by the expression $-\Delta + 4\pi^2 |t|^2$, or some scaled version of it.<fnref n="2" /> However, the Schrödinger representation $\rho_1$ of $H_n$ acting on $L^2(\mathbb{R}^n)$ and the associated Lie algebra representation, naturally acting on $\mathcal{S}(\mathbb{R}^n)$, clearly relate each of the realisations (R1)-(R3) to the others. Moreover, each of these realisations features a sum of squares. It ought therefore to be natural to assume that similar realisations should be available for the canonical harmonic oscillator on $H_n$.

The special role of the Heisenberg Lie algebra $\mathfrak{h}_n$ in this context is not coincidental: it is precisely the Lie algebra which is generated by the partial derivatives $\partial_{t_j}$ and the multiplication operators for the coordinate functions $t_k, j, k = 1, \dots, n$. It is well known that $\mathfrak{h}_n$ is stratified, therefore permits a (canonical) homogeneous structure, and that the sums of squares in the identities above are essentially related to the first stratum of $\mathfrak{h}_n$.

An operator on $H_n$ satisfying criteria analogous to (R1)-(R3) should clearly involve left-invariant (or alternatively right-invariant) vector fields on $H_n$, which are uniquely determined by some vectors in $\mathfrak{h}_n$, and a scalar potential expressed in terms of the coordinate functions on $H_n$. It ought therefore to be natural to study the Lie algebra generated by the standard basis of left-invariant vector fields, here denoted by $X_1, \dots, X_{2n+1}$, and the multiplication operators defined by the coordinates $t_1, \dots, t_{2n+1}$ on $\mathfrak{h}_n \cong \mathbb{R}^{2n+1}$ which determine the coordinates in which the vector fields are written. The resulting Lie algebra and its representation theory were first studied in Dynin [1], and in more detail in Folland [5]. This Lie algebra, which we shall call the Dynin-Folland Lie algebra, is in fact stratified and thus admits a sub-Laplacian. Endowed with the canonical homogeneous structure arising from the stratification, the Dynin-Folland Lie algebra together with its associated connected, simply 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. We provide a concrete formula for this operator and describe the asymptotic growth of its eigenvalues, using results by Mohamed, Lévy-Bruhl and Nourrigat [9].

<footnote n="1">¹extended to the universal enveloping algebra $u(\mathfrak{h}_n)$.</footnote>

<footnote n="2">²The factor $4\pi^2$ 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.</footnote>
## 2. The Dynin–Folland Group and its Representations

### 2.1. The Dynin–Folland Lie Algebra

In order to present our results, in this and the next sections we fix the notation and recall the fundamental results about the Dynin–Folland Lie algebra and group and its generic unitary irreducible representations due to Dynin [1] and Folland [5]. For more details we refer to [3, 5, 10].

We choose the usual exponential coordinates for the Heisenberg group $H_n$, thus express the group law by

$$ (t_{2n+1}, t_{2n}, \dots, t_1)(t'_{2n+1}, t'_{2n}, \dots, t'_1) = \left( t_{2n+1} + t'_{2n+1} + \frac{1}{2} \sum_{j=1}^{n} (t_j t'_{n+j} - t'_j t_{n+j}), t_{2n} + t'_{2n}, \dots, t_1 + t'_1 \right). $$

We can also group the variables as $\tilde{t}_3 := t_{2n+1}$, $\tilde{t}_2 := (t_{2n}, \dots, t_{n+1})$, $\tilde{t}_1 := (t_n, \dots, t_1)$ and rewrite the group law as

$$ (\tilde{t}_3, \tilde{t}_2, \tilde{t}_1)(\tilde{t}'_3, \tilde{t}'_2, \tilde{t}'_1) = \left( \tilde{t}_3 + \tilde{t}'_3 + \frac{1}{2} \left( \langle \tilde{t}_1, \tilde{t}'_2 \rangle - \langle \tilde{t}_2, \tilde{t}'_1 \rangle \right), \tilde{t}_2 + \tilde{t}'_2, \tilde{t}_1 + \tilde{t}'_1 \right). \quad (1) $$

In these coordinates, one can realise the Schrödinger representation $\rho_K$ of formal dimension $|K|^n$, $K \in \mathbb{R} \setminus \{0\}$, acting on $f \in L^2(\mathbb{R}^n)$, as

$$ (\rho_K(\tilde{t}_3, \tilde{t}_2, \tilde{t}_1)f)(\tilde{t}'_1) = e^{2\pi i \kappa (\tilde{t}_3 + \frac{1}{2}\langle \tilde{t}_1, \tilde{t}_2 \rangle + \langle \tilde{t}_2, \tilde{t}'_1 \rangle)} f(\tilde{t}'_1 + \tilde{t}_1). $$

The real Lie algebra $\tilde{\mathfrak{g}}$ of operators generated by the left-invariant vector fields on $H_n$

$$ X_j = \partial_{t_j} - \frac{1}{2} t_{n+j} \partial_{t_{2n+1}}, \quad X_{n+j} = \partial_{t_{n+j}} + \frac{1}{2} t_j \partial_{t_{2n+1}}, \quad X_{2n+1} = \partial_{t_{2n+1}}, \quad j = 1, \dots, n, $$

and by the multiplication operators $Y_k = 2\pi i t_k$ for $k = 1, \dots, 2n+1$ is 3-step nilpotent and, as a vector space, isomorphic to $\mathbb{R} \times \mathbb{R}^{2n+1} \times \mathfrak{h}_n \cong \mathbb{R}^{4n+3}$. If we denote by $Z$ the multiplication by the constant $2\pi i$ and identify the operators $Z, Y_1, \dots, X_{2n+1}$ with the standard basis vectors in $\mathbb{R}^{4n+3}$, the isomorphism is realised by equipping $\mathbb{R}^{4n+3}$ with the Lie bracket defined by

$$ [X_j, X_{n+j}] = X_{2n+1}, \quad [X_j, Y_{2n+1}] = -\frac{1}{2} Y_{n+j}, \quad [X_{n+j}, Y_{2n+1}] = \frac{1}{2} Y_j, \quad [X_k, Y_k] = Z, \quad (2) $$

for $j = 1, \dots, n$ and $k = 1, \dots, 2n+1$, and vanishing brackets otherwise. We will denote this Lie algebra by $\mathfrak{h}_{n,2}$ and refer to it as the Dynin–Folland Lie algebra. The connected, simply connected Lie group obtained by exponentiation will be referred to as the Dynin–Folland group and denoted by $H_{n,2}$. The Lie bracket relations immediately reveal that the Lie sub-algebra generated by $Z, Y_1, \dots, Y_{2n+1}$ is Abelian, and hence $H_{n,2}$ can be viewed as a semi-direct product of the form $\mathbb{R}^{2n+2} \times H_n$. Using exponential coordinates and identifying any element of $H_{n,2}$ with its corresponding coordinate vector $(z, y_1, \dots, y_{2n+1}, x_{2n+1}, \dots, x_1) := (z, y, x) \in \mathbb{R} \times \mathbb{R}^{2n+1} \times \mathbb{R}^{2n+1}$, the $H_{n,2}$-group law can be expressed by

$$ (z, y, x)(z', y', x') = \left( z + z' + \frac{1}{2} \left( \langle x, y' \rangle - \langle y, x' \rangle \right) - \frac{1}{8} \langle y - y', [x, x'] \rangle, \right. \\ \left. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad y + y' + \frac{1}{4} (\mathrm{ad}^*(x)y' - \mathrm{ad}^*(x')y), x \cdot x' \right) \quad (3) $$

provided we denote by $\langle x, y' \rangle$ the inner product $\sum_{j=1}^{2n+1} x_j y'_j$, by $[x, x']$ the $\mathfrak{h}_n$-Lie bracket of the vectors $\sum_{j=1}^{2n+1} x_j X_j$ and $\sum_{j=1}^{2n+1} x'_j X_j$ in $\mathfrak{h}_n$, by $x \cdot x'$ the $x''$-coordinates of $(0, 0, x'') = (0, 0, x) \cdot (0, 0, x') \in \mathbb{H} \le \mathbb{H}_{n,2}$, and by $\mathrm{ad}^*$ the coadjoint representation of $\mathfrak{h}_n \cong \mathbb{R}^{2n+1}$ on $\mathfrak{h}_n^* \cong \mathbb{R}^{2n+1}$ given by

$$ \mathrm{ad}^*(\tilde{t}_3, \tilde{t}_2, \tilde{t}_1)(\tilde{t}'_3, \tilde{t}'_2, \tilde{t}'_1) = (0, -\tilde{t}_3 \tilde{t}'_1, \tilde{t}_3 \tilde{t}'_2). $$
## 2.2. Stratification and Unitary Irreducible Representations

The Lie bracket (2) admits the stratification

$$ g_3 := \mathbb{R}Z, \quad g_2 := \mathbb{R}\text{-span}\{Y_1, \dots, Y_{2n}, X_{2n+1}\}, \quad g_1 := \mathbb{R}\text{-span}\{Y_{2n+1}, X_{2n}, \dots, X_1\}, $$

which possesses a canonical family of homogeneous dilations $\{D_r\}_{r>0}$ on $\mathfrak{h}_{n,2}$ given by

$$ \begin{aligned} D_r(Z) &= r^3 Z, \\ D_r(Y_k) &= r^2 Y_k, \qquad D_r(X_{2n+1}) = r^2 X_{2n+1}, \\ D_r(X_k) &= r X_k, \qquad D_r(Y_{2n+1}) = r Y_{2n+1}, \end{aligned} \qquad (4) $$

for $k=1, \dots, 2n$. The sub-Laplacian $\mathcal{L}_{\mathbf{H}_{n,2}}$ induced by the above stratification is the left-invariant differential operator on $\mathbf{H}_{n,2}$ corresponding to the sum of squares

$$ X_1^2 + \dots + X_{2n}^2 + Y_{2n+1}^2 \in \mathfrak{u}(\mathfrak{h}_{n,2}). $$

Moreover, for $l := \kappa Z^* \in \mathfrak{h}(g)^*$ with $\kappa \in \mathbb{R} \setminus \{0\}$ the matrix representation of the corresponding symplectic form $B_l = l([\cdot, \cdot])$ immediately reveals that $B_l$ is non-degenerate on $\mathfrak{h}_{n,2}/\mathfrak{h}(\mathfrak{h}_{n,2}) \times \mathfrak{h}_{n,2}/\mathfrak{h}(\mathfrak{h}_{n,2})$, i.e., up to Plancherel measure zero all unitary irreducible representations are square-integrable modulo the centre $Z(\mathbf{H}_{n,2})$, and its Pfaffian (characterizing the Plancherel measure) is given by $\mathrm{Pf}(l) = |\kappa|^{2n+1}$. These generic representations of $\mathbf{H}_{n,2}$, denoted by $\pi_\kappa$, $\kappa \in \mathbb{R} \setminus \{0\}$, can be induced by the characters $\chi_\kappa := e^{2\pi i \langle \kappa Z^*, \cdot \rangle}$ of the normal Abelian subgroup $\mathbb{R}^{2n+2} \le \mathbf{H}_{n,2}$: for a fixed $\kappa \in \mathbb{R} \setminus \{0\}$ the action of $\pi_\kappa$ on the representation space $\mathcal{H}_\kappa \cong L^2(\mathbf{H}_n)$ is explicitly given by

$$ (\pi_{\kappa}(z, y, x)f)(t) = e^{2\pi i \kappa z} e^{2\pi i \kappa \langle t \cdot \frac{1}{2}x, y \rangle} f(t \cdot x) \qquad (5) $$

for $f \in L^2(\mathbf{H}_n)$, where $t \cdot \frac{1}{2}x$ and $t \cdot x$ again denote the $\mathbf{H}_n$-group products of the corresponding coordinate vectors.

## 3. The Harmonic Oscillator on $\mathbf{H}_n$

The representation $\pi := \pi_1$ for $\kappa = 1$ defined in (5) was the object of interest in Dynin's account [1] since it served the purpose of introducing a Weyl quantization on $\mathbf{H}_n$. For our definition of the harmonic oscillator $\mathcal{D}_{\mathbf{H}_n}$ on $\mathbf{H}_n$ this representation plays the same crucial role as the Schrödinger representation does for $\mathcal{D}_{\mathbb{R}^n}$. For this reason the analogue of (R3) yields a canonical definition of $\mathcal{D}_{\mathbf{H}_n}$. The analogues of (R1) and (R2) will be an immediate consequence of our choice.

**Definition 1.** For the basis $\{Z, Y_1, \dots, X_{2n+1}\}$ of the Dynin–Folland Lie algebra $\mathfrak{h}_{n,2}$ and the representation $\pi = \pi_1 \in \hat{\mathbf{H}}_{n,2}$ realised on the representation space $L^2(\mathbf{H}_n)$, we define the harmonic oscillator on $\mathbf{H}_n$ to be the positive essentially self-adjoint operator

$$ \mathcal{D}_{\mathbf{H}_n} := d\pi(-\mathcal{L}_{\mathbf{H}_{n,2}}) = -d\pi(X_1)^2 - \dots - d\pi(X_{2n})^2 - d\pi(Y_{2n+1})^2, $$

whose natural domain includes the space of smooth vectors $\mathcal{H}_\pi^\infty \cong \mathcal{S}(\mathbf{H}_n)$.

The essentially self-adjoint differential operator $\mathcal{D}_{\mathbf{H}_n}$ on $\mathbf{H}_n$ admits the following three realisations:

(R1') the differential operator $-\mathcal{L}_{\mathbf{H}_n} + 4\pi^2 t_{2n+1}^2$;

(R2') the Dynin–Weyl quantization on $\mathbf{H}_n$ of the symbol $\sigma(t, \xi) := 4\pi^2(\xi_1^2 + \dots + \xi_{2n}^2 + t_{2n+1}^2)$ with $t, \xi \in \mathbb{R}^{2n+1}$;

(R3') the image $d\pi(-\mathcal{L}_{\mathbf{H}_{n,2}})$ of the negative sub-Laplacian $-\mathcal{L}_{\mathbf{H}_{n,2}}$ under the infinitesimal representation $d\pi$ of the Dynin–Folland Lie algebra $\mathfrak{h}_{n,2}$.
Since the Lie algebra isomorphism $\mathfrak{h}_{n,2} \to \tilde{\mathfrak{g}}$ defined by (2) is precisely $d\pi$, we immediately have

$$ \begin{aligned} \mathcal{Q}_{\mathbf{H}_n} &= -\mathcal{L}_{\mathbf{H}_n} + 4\pi^2 t_{2n+1}^2 \\ &= -\left(\partial_{t_1} - \frac{1}{2}t_{n+1}\partial_{t_{2n+1}}\right)^2 - \cdots - \left(\partial_{t_{2n}} + \frac{1}{2}t_n\partial_{t_{2n+1}}\right)^2 + 4\pi^2 t_{2n+1}^2, \end{aligned} \quad (6) $$

thus (R1'). As for Dynin's Weyl quanization of $\sigma(t, \xi) := 4\pi^2(\xi_1^2 + \cdots + \xi_{2n}^2 + t_{2n+1}^2)$, it suffices to recall that every monomial $(2\pi i \xi_j)^m, j = 1, \dots, 2n+1, m \in \mathbb{N}$, is mapped to the left-invariant differential operator $X_i^m$, and that multiplication by the monomial $(2\pi i t_k)^m, k = 1, \dots, 2n+1$, is mapped to the multiplication operator $Y_k^m = (2\pi i t_k)^m$. For more details we refer to [1] and [10, § 5].

Note that equivalently we could realise $\pi$ as a direct summand of the left regular representation of $\mathbf{H}_{n,2}$, thereby replacing the left-invariant sub-Laplacian $\mathcal{L}_{\mathbf{H}_n}$ in (6) by the right-invariant one. The spectral asymptotics, however, would not change (cf. [9]).

## 4. Spectral Properties

The harmonic oscillator $\mathcal{Q}_{\mathbf{H}_n}$ has purely discrete spectrum in $(0, \infty)$ and we obtain the asymptotic growth rate of its eigenvalues by employing a powerful method, which in the setting of stratified groups was developed in Mohamed, Lévy-Bruhl and Nourrigat [9] (see also [8] for a short version in English), and which was extended to the setting of graded groups in ter Elst and Robinson [2]. If $G$ is nilpotent but non-Abelian, then for almost every unitary irreducible representation $\pi \in \hat{G}$ the representation space $\mathcal{H}_\pi$ is infinite-dimensional. Moreover, if $G$ is graded, then a homogeneous left-invariant differential operator $\mathcal{R}$ on $G$ is said to be a Rockland operator if for every non-trivial $\pi \in \hat{G}$ the operator $d\pi(\mathcal{R})$ is injective on the space of smooth vectors $\mathcal{H}_\pi^\infty \subset \mathcal{H}_\pi$. This is the case if and only if the operator $\mathcal{R}$ is hypoelliptic. The equivalence was conjectured by Rockland and settled in Helffer-Nourrigat [6]. Hulanicki, Jenkins and Ludwig [7] showed that if $\mathcal{R}$ is positive, then for every $\pi \in \hat{G}$ the operator $d\pi(\mathcal{R})$ has purely discrete spectrum in $(0, \infty)$. In particular, every negative sub-Laplacian $-\mathcal{L}$ on a stratified group is a positive Rockland operator. A concrete description of the spectrum of $-d\pi(\mathcal{L})$ is due to [9], and for general $d\pi(\mathcal{R})$ due to [2]: in both cases the authors showed that the number of eigenvalues of a given $-d\pi(\mathcal{L})$ or $d\pi(\mathcal{R})$, counted with multiplicities, asymptotically grows like the volumes of certain subsets of the corresponding coadjoint orbit $\mathcal{O}_\pi$. The subsets in question are determined (up to a multiplicative constant) by a (any) homogeneous quasi-norm on $\mathfrak{g}^*$. These estimates also give an asymptotic value for the magnitude of a given eigenvalue.

In the case of $G = \mathbf{H}_{n,2}$ and $\mathcal{R} = -\mathcal{L}_{\mathbf{H}_{n,2}}$, the realisation of $\pi = \pi_1 \in \hat{\mathbf{H}}_{n,2}$ in $\mathcal{H}_\pi = L^2(\mathbf{H}_n)$ given by (5) makes these results readily available for $d\pi(\mathcal{R}) = \mathcal{Q}_{\mathbf{H}_n}$. The choice of a convenient quasi-norm on $\mathfrak{h}_{n,2}^*$ and the fact that the coadjoint orbit $\mathcal{O}_\pi$ is flat facilitate the computation of the volumes in question substantially. One can use this to show:

**Theorem.** The harmonic oscillator $\mathcal{Q}_{\mathbf{H}_n}$ on the Heisenberg group $\mathbf{H}_n$ has a purely discrete spectrum $\text{spec}(\mathcal{Q}_{\mathbf{H}_n}) \subset (0, \infty)$. 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

$$ \lambda_k \sim k^{\frac{2}{6n+3}} \text{ for } k = 1, 2, \dots $$

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 the homogeneous dimension of the first two strata of $\mathfrak{h}_{n,2}$, i.e., of the subspace
$g_1 \oplus g_2 \subseteq \mathfrak{h}_{n,2}$, while the denominator 2 is the homogeneous degree of $-\mathcal{L}_{\mathbb{H}_{n,2}}$. For a proof of the spectral asymptotics we refer to the preprint [11], especially the proof of Proposition 6.3. The eigenvectors are clearly elements of $\bigcap_{l=1}^{\infty} \text{dom}(\overline{\mathcal{Q}_{\mathbb{H}_n}})^l) \subseteq L^2(\mathbb{H}_n)$; since $\bigcap_{l=1}^{\infty} \text{dom}(\overline{d\pi(\mathcal{R})})^l) = \mathcal{H}_{\pi}^{\infty}$ (see [2, Prop. 2.1]), this set coincides with $\mathcal{S}(\mathbb{H}_n)$. The eigenvectors form an orthonormal basis of $L^2(\mathbb{H}_n)$ because $\mathcal{Q}_{\mathbb{H}_n}$ possesses a compact resolvent as was shown for general $d\pi(\mathcal{R})$ in [6].

References

[1] A. S. Dynin, "Pseudodifferential operators on the Heisenberg group", Dokl. Akad. Nauk SSSR **225** (1975), p. 1245-1248.

[2] A. F. M. ter Elst, D. W. Robinson, "Spectral estimates for positive Rockland operators", in Algebraic groups and Lie groups, Australian Mathematical Society Lecture Series, vol. 9, Cambridge University Press, 1997, p. 195-213.

[3] V. Fischer, D. Rottensteiner, M. Ruzhansky, "Heisenberg-Modulation Space on the Crossroads of Coorbit Theory and Decomposition Space Theory", https://arxiv.org/abs/1812.07876, 2018.

[4] G. B. Folland, *Harmonic Analysis in Phase Space*, Princeton University Press, 1989.

[5] ——, "Meta-Heisenberg groups", in *Fourier analysis (Orono, ME, 1992)*, Lecture Notes in Pure and Applied Mathematics, vol. 157, Marcel Dekker, 1994, p. 121-147.

[6] B. Helffer, J. Nourrigat, "Caractérisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué", *Commun. Partial Differ. Equations* **4** (1979), no. 8, p. 899-958.

[7] A. Hulanicki, J. W. Jenkins, J. Ludwig, "Minimum eigenvalues for positive, Rockland operators", *Proc. Am. Math. Soc.* **94** (1985), no. 4, p. 718-720.

[8] A. Morame, P. Lévy-Bruhl, J. Nourrigat, "Spectral theory and representations of nilpotent groups", *Bull. Am. Math. Soc.* **26** (1992), no. 2, p. 299-303.

[9] ——, "Étude spectrale d'opérateurs liés à des représentations de groupes nilpotents", *J. Funct. Anal.* **113** (1993), no. 1, p. 65-93.

[10] D. Rottensteiner, "Time-Frequency Analysis on the Heisenberg Group", PhD Thesis, Imperial College London, 2014, https://spiral.imperial.ac.uk/handle/10044/1/24568, 154 pages.

[11] D. Rottensteiner, M. Ruzhansky, "Harmonic and anharmonic oscillators on the Heisenberg group", https://arxiv.org/abs/1812.09620, 2018.

[12] E. M. Stein, *Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals*, Princeton Mathematical Series, vol. 43, Princeton University Press, 1993, with the assistance of Timothy S. Murphy, xiv+695 pages.