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We investigate Weyl type asymptotics of functional-difference operatorsassociated to mirror curves of special del Pezzo Calabi-Yau threefolds. Theseoperators are $H(\zeta)=U+U^{-1}+V+\zeta V^{-1}$ and$H_{m,n}=U+V+q^{-mn}U^{-m}V^{-n}$, where $U$ and $V$ are self-adjoint Weyloperators satisfying $UV=q^{2}VU$ with $q=e^{i\pi b^{2}}$, $b>0$ and $\zeta>0$,$m,n\in\mathbb{N}$. We prove that $H(\zeta)$ and $H_{m,n}$ are self-adjointoperators with purely discrete spectrum on $L^{2}(\mathbb{R})$. Using thecoherent state transform we find the asymptotical behaviour for the Riesz mean$\sum_{j\ge 1}(\lambda-\lambda_{j})_{+}$ as $\lambda\to\infty$ and prove theWeyl law for the eigenvalue counting function $N(\lambda)$ for these operators,which imply that their inverses are of trace class.

Weyl type asymptotics and bounds for the eigenvalues of functional-difference operators for mirror curves