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

TL;DR

This paper establishes Weyl asymptotics and eigenvalue bounds for certain functional-difference operators linked to mirror curves of del Pezzo Calabi-Yau threefolds, demonstrating their spectral properties and trace class nature.

Contribution

It proves self-adjointness, discreteness of spectrum, and Weyl law for eigenvalues of these specialized operators, extending spectral theory in mathematical physics.

Findings

Operators are self-adjoint with purely discrete spectrum.

Weyl law established for eigenvalue counting function.

Inverses of operators are trace class.

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