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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 $H(ζ)=U+U^{-1}+V+ζV^{-1}$ 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 $q=e^{iπb^{2}}$, $b>0$ and $ζ>0$, $m,n\in\mathbb{N}$. We prove that $H(ζ)$ 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 $\sum_{j\ge 1}(λ-λ_{j})_{+}$ as $λ\to\infty$ and prove the Weyl law for the eigenvalue counting function $N(λ)$ for these operators, which imply that their inverses are of trace class.