Local solvability of linear differential operators with double characteristics II:: Sufficient conditions for left-invariant differential operators of order two on the Heisenberg group

Consider second order differential operators on the Heisenberg group of the form L = Sigma(2n)(j,k=1)alpha jkVjVk + i alpha U, where the Vj are the usual left-invariant vector fields on the Heisenberg group satisfying the commutation relations [V-j, Vj+n] = U for 1 <= j <= n, and where A = (alpha(jk)) is a complex symmetric matrix. It has been shown in the first article of this series that the operator L can be locally solvable only, if it is essential dissipative, i.e., if there exists some theta epsilon R such that Re(ei theta A) >= 0 (provided the real and imaginary part of A satisfy a mild rank condition and the Hamiltonians associated to the real part of A, the imaginary part and their commutator are linearly independent). Assuming therefore that A has semi-definite real part, we shall prove local solvability for large classes of such operators L. Our results widely extend all results on the case of a non-real, semi-definite coefficient matrix A known to date, in particular those obtained in joint work with F. Ricci under the hypothesis that L satisfies Sjostrand's cone condition. Already those results had shown that local solvability does hold "generically", if L is essentially dissipative. More precise sufficient conditions for local solvability involve, however, spectral properties of the Hamiltonian associated to A and properties of the subprincipal symbol (which for our operators essentially just means the coefficient ce) in an intricate way. Our results are achieved by showing that an integration by parts argument, which had been applied in different forms in previous articles, ultimately allows for a reduction to the case of operators L whose associated Hamiltonian has a purely real spectrum. Various examples are given in order to indicate the potential scope of this approach and to illuminate some further conditions that will be introduced in the article.