Equations of evolution on the Heisenberg group. II (Q1326733)

[For Part I cf. ibid. 32, 749-761 (1992; Zbl 0801.22003).] Suppose that the hyperbolic operators of higher order on the Heisenberg group \(H^n\) have the form \(P = \partial^m_t + \sum^m_{j = 1} a_j \partial_t^{m - j}\) where \(a_j\) are homogeneous right invariant differential operators of order \(j\) on \(\mathbb{H}^n\). This paper investigates the relation between the hyperbolicity of the operator \[ \pi \bigl( P(\zeta) \bigr) = (i \zeta)^m + \sum^m_{j = 1} \pi (a_j) (i \zeta)^{m - j} \] for any nontrivial irreducible unitary representation \(\pi\) on \(\mathbb{H}^n\) and the well-posedness of the Cauchy problem for \(P\). The main result here asserts that if the Fourier transform of \(i^mP^\delta\) is strictly hyperbolic of nondegenerate type and \(\pi (P (\zeta))\) is of hyperbolic type and satisfies some commutativity relations among its coefficients for any \(\pi\), then the Cauchy problem for \(P\) is solvable at the origin.