Geometric Results for a Class of Hyperbolic Operators with Double Characteristics ,II.

Let p be the principal symbol of a hyperbolic (pseudo) differential operator of order m admitting at most double characteristic roots. Suppose that at each point ρ of the double characteristic manifold Σ of p the Hamiltonian matrix of p, Fp, hasa Jordan block of dimension 4. We prove a necessary and sufficient condition on p in order that its bicharacteristic curves have limit points belonging to Σ. It is shown that if no bicharacteristic curve of p has a limit point belonging to Σ then the Cauchy problem for p is well-posed, provided the usual Levi conditions on thelower order terms are satisfied.