Let $U \subset \mathbb{R}^n$ be an open coordinate domain, let $m \in \mathbb{N}$, and let $0 < h \leq h_0$. For each multi-index $\alpha \in \mathbb{N}_0^n$ with $|\alpha| \leq m$, let
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\begin{align*}
a_\alpha: U \times [0,h_0] \to \mathbb{C}
\end{align*}
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be smooth in the $x$-variable, continuous at $h = 0$, and uniformly bounded with all $x$-derivatives for $0 \leq h \leq h_0$. Define the semiclassical differential operator
If $P_h$ is a globally defined semiclassical differential operator of order $m$ on a smooth manifold $M$, and its top-order coefficients define a symmetric contravariant $m$-tensor at $h=0$, then the local expressions above agree under coordinate changes. Consequently $p_m$ defines a globally well-defined smooth function