  Although there is no canonical version of the harmonic oscillator on the
Heisenberg group $\mathbf{H}_n$ so far, we make a strong case for a particular
choice of operator by using the representation theory of the Dynin-Folland
group $\mathbf{H}_{n, 2}$, a $3$-step stratified Lie group, whose generic
representations act on $L^2(\mathbf{H}_n)$. Our approach is inspired by the
connection between the harmonic oscillator on $\mathbb{R}^n$ and the sum of
squares in the first stratum of $\mathbf{H}_n$ in the sense that we define the
harmonic oscillator on $\mathbf{H}_n$ as the image of the sub-Laplacian
$\mathcal{L}_{\mathbf{H}_{n, 2}}$ under the generic unitary irreducible
representation $\pi$ of the Dynin-Folland group which has formal dimension
$d_\pi = 1$. This approach, more generally, permits us to define a large class
of so-called anharmonic oscillators by employing positive Rockland operators on
$\mathbf{H}_{n, 2}$. By using the methods developed in ter Elst and Robinson
[tERo], we obtain spectral estimates for the harmonic and anharmonic
oscillators on $\mathbf{H}_n$. Moreover, we show that our approach extends to
graded $SI/Z$-groups of central dimension $1$, i.e., graded groups which
possess unitary irreducible representations which are square-integrable modulo
the $1$-dimensional center $Z(G)$. The latter part of the article is concerned
with spectral multipliers. By combining ter Elst and Robinson's techniques with
recent results in [AkRu18], we obtain useful
$L^\mathbf{p}$-$L^\mathbf{q}$-estimates for spectral multipliers of the
sub-Laplacian $\mathcal{L}_{\mathbf{H}_{n, 2}}$ and, in fact more generally, of
general Rockland operators on general graded groups. As a by-product, we
recover the Sobolev embeddings on graded groups established in [FiRu17], and
obtain explicit hypoelliptic heat semigroup estimates.
