  We extend the notions of CR GJMS operators and Q-curvature to the case of
partially integrable CR structures. The total integral of the CR Q-curvature
turns out to be a global invariant of compact nondegenerate partially
integrable CR manifolds equipped with an orientation of the bundle of contact
forms, which is nontrivial in dimension at least five. It is shown that its
variation is given by the curvature-type quantity called the CR obstruction
tensor, which is introduced in the author's previous work. Moreover, we
consider the linearized CR obstruction operator. Based on a
scattering-theoretic characterization, we discuss its relation to the CR
deformation complex of integrable CR manifolds. The same characterization is
also used to determine the Heisenberg principal symbol of the linearized CR
obstruction operator.
