We study the question of local solvability for second-order, left-invariant differential operators on the Heisenberg group
H
n
\mathbb {H}_n
, of the form
\[
P
Λ
=
∑
i
,
j
=
1
n
λ
i
j
X
i
Y
j
=
t
X
Λ
Y
,
\mathcal {P}_\Lambda = \sum _{i,j=1}^{n} \lambda _{ij}X_i Y_j={\,}^t X\Lambda Y,
\]
where
Λ
=
(
λ
i
j
)
\Lambda =(\lambda _{ij})
is a complex
n
×
n
n\times n
matrix. Such operators never satisfy a cone condition in the sense of Sjöstrand and Hörmander. We may assume that
P
Λ
\mathcal {P}_\Lambda
cannot be viewed as a differential operator on a lower-dimensional Heisenberg group. Under the mild condition that
Re
Λ
,
\operatorname {Re}\Lambda ,
Im
Λ
\operatorname {Im}\Lambda
and their commutator are linearly independent, we show that
P
Λ
\mathcal {P}_\Lambda
is not locally solvable, even in the presence of lower-order terms, provided that
n
≥
7
n\ge 7
. In the case
n
=
3
n=3
we show that there are some operators of the form described above that are locally solvable. This result extends to the Heisenberg group
H
3
\mathbb {H}_3
a phenomenon first observed by Karadzhov and Müller in the case of
H
2
.
\mathbb {H}_2.
It is interesting to notice that the analysis of the exceptional operators for the case
n
=
3
n=3
turns out to be more elementary than in the case
n
=
2.
n=2.
When
3
≤
n
≤
6
3\le n\le 6
the analysis of these operators seems to become quite complex, from a technical point of view, and it remains open at this time.