Weyl Type Asymptotics and Bounds for the Eigenvalues of Functional-Difference Operators for Mirror Curves

Abstract

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 ${q = {\rm e}^{{\rm i}\pi b^{2}}}$, ${b > 0}$ 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 ${\lambda \to \infty}$ and prove the Weyl law for the eigenvalue counting function ${N(\lambda)}$ for these operators, which imply that their inverses are of trace class.