Let $M$ be a smooth $n$-dimensional manifold, and let $\{(U_j,\kappa_j)\}_{j\in J}$ be a locally finite coordinate cover of $M$, where $\kappa_j:U_j\to V_j\subset\mathbb{R}^n$ is a smooth chart. Let $\{\chi_j\}_{j\in J}$ be a locally finite smooth partition of unity on $M$ with $\chi_j\in C_c^\infty(U_j)$, and for each $j\in J$ choose $\psi_j\in C_c^\infty(U_j)$ such that $\psi_j=1$ on an open neighbourhood of $\operatorname{supp}\chi_j$.

For each $j\in J$, let $a_j\in S_h^m(T^*U_j)$ be a semiclassical symbol, interpreted in the chart $\kappa_j$, and assume that the family of local symbol seminorms of the $a_j$ is uniformly bounded on compact subsets of $T^*M$ after passage to the charts. Assume also that the principal symbol classes

\begin{align*} \sigma_j := a_j \bmod hS_h^{m-1}(T^*U_j) \end{align*}

are compatible on overlaps: for every $j,k\in J$, the two classes $\sigma_j$ and $\sigma_k$ agree on $T^*(U_j\cap U_k)$ after applying the cotangent coordinate change induced by $\kappa_k\circ\kappa_j^{-1}$.

For $h\in(0,h_0]$, define

\begin{align*} A_h := \sum_{j\in J} \chi_j\,\operatorname{Op}_{h,\kappa_j}(a_j)\,\psi_j, \end{align*}

where multiplication by $\chi_j$ and $\psi_j$ is understood on $M$. Suppose the Schwartz kernel of this locally finite sum has proper projections to both factors of $M\times M$; equivalently, for every compact set $K\subset M$, only finitely many summand kernels meet $K\times M$ or $M\times K$, and the corresponding intersections project to compact subsets of $M$.

Then $A_h$ defines a properly supported semiclassical pseudodifferential operator of order $m$ on $M$:

\begin{align*} A_h\in \Psi_h^m(M). \end{align*}

Its principal symbol is the unique global class

\begin{align*} \sigma(A_h)\in S_h^m(T^*M)/hS_h^{m-1}(T^*M) \end{align*}

whose restriction to $T^*U_j$ is $\sigma_j$ for every $j\in J$.