Let $M$ be a compact smooth manifold, let $h_0>0$ be fixed, and let $h \in (0,h_0]$ be the semiclassical parameter. For each $m \in \mathbb{R}$, let $\Psi_h^m(M)$ denote the semiclassical pseudodifferential operators of order $m$ on $M$, and let $S^m(T^*M)$ denote the corresponding semiclassical symbol class on the cotangent bundle. Use the semiclassical principal-symbol convention in which two order $m$ symbols define the same principal symbol if their difference lies in $hS^{m-1}(T^*M)$. Residual operators $\Psi_h^{-\infty}(M)$ are understood in the standard semiclassical sense: smoothing operators whose full symbols and Schwartz kernels are $O(h^N)$ in all seminorms for every $N \in \mathbb{N}$.

Then there is a well-defined principal symbol map

\begin{align*} \sigma_h^m: \Psi_h^m(M) \to S^m(T^*M) / hS^{m-1}(T^*M) \end{align*}

whose kernel is exactly $h\Psi_h^{m-1}(M)$. Equivalently, the sequence

\begin{align*} 0 \to h\Psi_h^{m-1}(M) \to \Psi_h^m(M) \xrightarrow{\sigma_h^m} S^m(T^*M) / hS^{m-1}(T^*M) \to 0 \end{align*}

is exact, where the first arrow is the inclusion map.