Riesz-means bounds for functional-difference operators for mirror curves

TL;DR

This paper estimates the asymptotic behavior of Riesz means for eigenvalues of a specific quantum operator linked to mirror curves in Calabi-Yau threefolds, extending spectral analysis techniques to this geometric context.

Contribution

It provides new bounds for Riesz means of eigenvalues of a quantum operator associated with mirror curves, connecting spectral analysis with geometric quantization.

Findings

Derived asymptotic estimates for Riesz means as eigenvalues grow large

Connected spectral properties of the operator to geometric structures in Calabi-Yau spaces

Extended spectral analysis methods to a new class of quantum operators

Abstract

Let \( P \) and \( Q \) be the quantum-mechanical momentum and position operators on \( L^2(\R) \). Let $\zeta>0.$ We provide estimates for the {\it Riesz means} $\varkappa(\lambda)$ associated with the system of eigenvalues of the operator \begin{align} H(\zeta) = \e^{-bP} + \e^{bP} + \e^{2\pi b Q} + \zeta \e^{-2\pi b Q} = U + U^{-1} + V + \zeta V^{-1}, \end{align} when $\lambda\rightarrow\infty.$ This operator arises in the quantisation of the local {\it del Pezzo Calabi-Yau threefold}, defined as the total space of the anti-canonical bundle over the {\it Hirzebruch surface} \( S = \mathbb{P}^{1} \times \mathbb{P}^{1} \). Our approach is motivated by the spectral analysis of $\varkappa(\lambda)$ in the framework developed by Laptev, Schimmer and Takhtajan in [13].