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

Let $P$ and $Q$ be the quantum-mechanical momentum and position operators on $L^2(\R)$. 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].