Analytic hypoellipticity for $\square_b + c$ on the Heisenberg group: an $L^2$ approach

In an interesting note, E.M. Stein observed some 20 years ago that while the Kohn Laplacian $\square_b$ on functions is neither locally solvable nor (analytic) hypoelliptic, the addition of a non-zero complex constant reversed these conclusions at least on the Heisenberg group, and Kwon reproved and generalized this result using the method of concatenations. Recently Hanges and Cordaro have studied this situation on the Heisenberg group in detail. Here we give a purely $L^2$ proof of Stein's result using the author's now classical construction of $(T^p)_\phi = \phi T^p +...,$ where $T$ is the 'missing direction' on the Heisenberg group.