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

We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators 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 Weyl operators 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-adjoint operators with purely discrete spectrum on $L^2(\mathbb{R})$. Using the coherent state transform we find the asymptotical behaviour for the Riesz mean ${\sum }_{j\ge 1}(\lambda -{\lambda }_j{)}_{+}$ as $\lambda \to \mathrm{\infty }$ and prove the Weyl law for the eigenvalue counting function $N(\lambda )$ for these operators, which imply that their inverses are of trace class.