|
|
|
Solvability of dissipative second order left-invariant differential operators on the Heisenberg group
We prove local solvability for large classes of operators of the form L = ∑j,k=12n ajk Vj Vk + i α U, where the Vj are left-invariant vector fields on the Heisenberg group satisfying the commutation relations [Vj,Vj+n] = U for 1 <= j <= n, and where A = (ajk) is a complex symmetric matrix with semi-definite real part. Our results widely extend all of the results for the case of non-real, semi-definite matrices A known to date, in particular those obtained recently jointly with F. Ricci under Sjöstrand's cone condition. They 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. Electronic Print: Front for the Mathematics ArXiv
Mathematics Subject Classification (1991):
35A05, 43A80
Bibliographical note: erscheint in J. Reine u. angew. Math.
|