$m\in C^\infty ({\bf R}^{2n};[1,+\infty
[)$ be an order function (see [@DiSj99] for more details about the pseudodifferential calculus) in the sense that for some $C_0,N_0>0$, [$$\label{in.1}
m(\rho )\le C_0\langle \rho -\mu \rangle^{N_0}m(\mu ),\ \rho ,\mu \in
{\bf R}^{2n},$$]{} where $\langle \rho -\mu \rangle= (1+|\rho -\mu |^2)^{1/2}$. Let $P=P(x,\xi ;h)\in S(m)$, meaning that $P$ is smooth in $x,\xi $ and satisfies [$$\label{in.2}
|\partial _{x,\xi }^\alpha P(x,\xi ;h)|\le C_\alpha m(x,\xi ),\ (x,\xi
)\in {\bf R}^{2n},\, \alpha
\in {\bf N}^{2n},$$]{} where $C_\alpha $ is independent of $h$. We also assume that [$$\label{in.3}
P(x,\xi ;h)\sim p_0(x,\xi )+hp_1(x,\xi )+...,\hbox{ in }S(m),$$]{} and write $p=p_0$ for the principal symbol. We impose the ellipticity assumption [$$\label{in.4}
\exists w\in {\bf C},\, C>0,\hbox{ such that }|p(\rho )-w|\ge m(\rho )/C,\
\forall \rho \in {\bf R}^{2n}.$$]{} In this case we let [$$\label{in.5}
P=P^w(x,hD_x;h)=\mathrm{Op}(P(x,h\xi ;h))$$]{} be the Weyl quantization of the symbol $P(x,h\xi ;h)$ that we can view as a closed unbounded operator on $L^2({\bf R}^n)$.

In the second case when $X$ is compact manifold, we let $P\in
S^m_{1,0}(T^*X)$ (the classical Hörmander symbol space ) of order $m> 0$, meaning that [$$\label{in.6}
|\partial _x^\alpha \partial _{\xi }^\beta P(x,\xi ;h)|\le C_{\alpha
  ,\beta }\langle \xi \rangle^{m-|\beta |},\ (x,\xi )\in T^*X,$$]{} where $C_{\alpha ,\beta }$ 