  The critical CR GJMS operator on a strictly pseudoconvex CR manifold is a
non-hypoelliptic CR invariant differential operator. We prove that, under the
embeddability assumption, it is essentially self-adjoint and has closed range.
Moreover, its spectrum is discrete, and the eigenspace corresponding to each
non-zero eigenvalue is a finite-dimensional subspace of the space of smooth
functions. As an application, we obtain a necessary and sufficient condition
for the existence of a contact form with zero CR $Q$-curvature.
