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

Let $P$ and $Q$ be the quantum-mechanical momentum and position operators on $L^2(\mathbb{R})$. Let $\zeta >0.$ We provide estimates for the {\it Riesz means} $\mathrm{\varkappa }(\lambda )$ associated with the system of eigenvalues of the operator \begin{align} H(ζ) = \e^{-bP} + \e^{bP} + \e^{2πb Q} + ζ\e^{-2πb Q} = U + U^{-1} + V + ζV^{-1}, \end{align} when $\lambda \to \mathrm{\infty }\mathrm{.}$ 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 $\mathrm{\varkappa }(\lambda )$ in the framework developed by Laptev, Schimmer and Takhtajan in [13].