Let $n \in \mathbb{N}$, let $h > 0$, let $m \in \mathbb{R}$, and identify $T^*\mathbb{R}^n$ with $\mathbb{R}^n_x \times \mathbb{R}^n_\xi$. Let $a:T^*\mathbb{R}^n \to \mathbb{C}$ be a complex-valued semiclassical symbol with $a \in S^m(T^*\mathbb{R}^n)$, and let $\overline a:T^*\mathbb{R}^n \to \mathbb{C}$ denote its pointwise complex conjugate. Define the semiclassical Weyl quantization $\operatorname{Op}_h^w(a): \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)$ by the oscillatory integral

\begin{align*} (\operatorname{Op}_h^w(a)u)(x) = (2\pi h)^{-n} \operatorname{Os}\!\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} e^{i(x-y)\cdot \xi/h} a\!\left(\frac{x+y}{2},\xi\right) u(y) \, d\mathcal{L}^n(y)\, d\mathcal{L}^n(\xi) \end{align*}

for $u \in \mathcal{S}(\mathbb{R}^n)$ and $x \in \mathbb{R}^n$. Equip $L^2(\mathbb{R}^n)$ with the inner product linear in the first entry and conjugate-linear in the second entry:

\begin{align*} (f,g)_{L^2(\mathbb{R}^n)}=\int_{\mathbb{R}^n} f(x)\overline{g(x)}\, d\mathcal{L}^n(x). \end{align*}

Then the formal adjoint of $\operatorname{Op}_h^w(a)$ is $\operatorname{Op}_h^w(\overline{a})$ in the sense that, for every $u,v \in \mathcal{S}(\mathbb{R}^n)$,

\begin{align*} (\operatorname{Op}_h^w(a)u,v)_{L^2(\mathbb{R}^n)} = (u,\operatorname{Op}_h^w(\overline{a})v)_{L^2(\mathbb{R}^n)}. \end{align*}

Equivalently,

\begin{align*} (\operatorname{Op}_h^w(a))^*=\operatorname{Op}_h^w(\overline{a}) \end{align*}

as formal adjoints of continuous maps $\mathcal{S}(\mathbb{R}^n)\to \mathcal{S}'(\mathbb{R}^n)$. In particular, if $a$ is real-valued, then $\operatorname{Op}_h^w(a)$ is formally self-adjoint.