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 $ζ>0.$ We provide estimates for the {\it Riesz means} $\varkappa(λ)$ 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 $λ\rightarrow\infty.$ 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 $\varkappa(λ)$ in the framework developed by Laptev, Schimmer and Takhtajan in [13].