be a properly supported scalar pseudodifferential operator in $\Psi^m(U)$ with full symbol
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\begin{align*}
a: U \times \mathbb{R}^n \to \mathbb{C}
\end{align*}
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in a fixed quantization convention. Assume that $a \in S^m(U \times \mathbb{R}^n;\mathbb{C})$ is elliptic, meaning that for every compact set $K \subset U$ there exist constants $C_K > 0$ and $R_K > 0$ such that
for all $x \in K$ and all $\xi \in \mathbb{R}^n$ with $|\xi| \geq R_K$. Assume moreover that this ellipticity is implemented by a global high-frequency reciprocal: there exists a smooth cutoff
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\begin{align*}
\chi: U \times \mathbb{R}^n \to [0,1]
\end{align*}
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such that $\chi(x,\xi)=0$ on a neighbourhood of the set where $a(x,\xi)=0$, such that $\chi-1 \in S^{-\infty}(U \times \mathbb{R}^n;\mathbb{C})$, and such that the map
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\begin{align*}
q_0: U \times \mathbb{R}^n \to \mathbb{C}
\end{align*}
\begin{align*}
QP - I \in \Psi^{-\infty}(U)
\end{align*}
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and
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\begin{align*}
PQ - I \in \Psi^{-\infty}(U).
\end{align*}
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Here $I:C_c^\infty(U)\to\mathcal{D}'(U)$ is the canonical inclusion operator. If $P$ is classical with principal homogeneous symbol
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\begin{align*}
a_m: U \times (\mathbb{R}^n\setminus\{0\}) \to \mathbb{C}
\end{align*}
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and the reciprocal cutoff $q_0$ is classical modulo $S^{-m-1}$ with leading homogeneous term $a_m^{-1}$, then $Q$ may be chosen classical with principal homogeneous symbol $a_m^{-1}$. In the general nonclassical case, the class of the full symbol of $Q$ in $S^{-m}/S^{-m-1}$ is represented by the high-frequency reciprocal $q_0$.