are independent of $h$. We also assume that we have an expansion as in (\[in.3\]), now in the sense that [$$\label{in.7} P(x,\xi ;h)-\sum_0^{N-1}h^jp_j(x,\xi )\in h^NS^{m-N}_{1,0}(T^*X),\ N=1,2,...$$]{} and we quantize the symbol $P(x,h\xi ;h)$ in the standard (non-unique) way, by doing it for various local coordinates and paste the quantizations together by means of a partition of unity. In the case $m>0$ we impose the ellipticity condition [$$\label{in.8} \exists C>0,\hbox{ such that }|p(x,\xi )|\ge \frac{\langle
  \xi \rangle^m}{C},\ |\xi |\ge C.$$]{}

Let $\Sigma (p)=\overline{p^*(T^*X)}$ and let $\Sigma _\infty
(p)$ be the set of accumulation points of $p(\rho _j)$ for all sequences $\rho _j\in T^*X$, $j=1,2,3,..$ that tend to infinity. The following theorem is a partial improvement of corresponding results in [@DeSjZw].

\[in1\] We adopt the general assumptions above. Let $z_0\in \partial \Sigma
(p)\setminus \Sigma _\infty (p)$ and assume that $dp\ne 0$ at every point of $p^{-1}(z_0)$. Then for every such point $\rho $ there exists $\theta \in {\bf R}$ (unique up to a multiple of $\pi $) such that $d(e^{-i\theta }(p-z_0))$ is real at $\rho $. We write $\theta
=\theta (\rho )$. Consider the following two cases:

-   \(A) For every $\rho \in p^{-1}(z_0)$, the maximal integral curve of $H_{\Re (e^{-i\theta (\rho 