A simple proof of the existence of tangent bicharacteristics for noneffectively hyperbolic operators

Abstract

The behavior of orbits of the Hamilton vector field $H_p$ of the principal symbol $p$ of a second-order hyperbolic differential operator is discussed. In our previous paper, assuming that $p$ is noneffectively hyperbolic on the doubly characteristic manifold $\Sigma$ of $p$, we have proved that if $H_S^{3}p=0$ on $\Sigma$ with the Hamilton vector field $H_S$ of some specified $S$, then there exists a bicharacteristic landing on $\Sigma$ tangentially. The aim of this paper is to provide a much more simple proof of this result since the previous proof was fairly long and rather complicated.