Weyl Type Asymptotics and Bounds for the Eigenvalues of Functional-Difference Operators for Mirror Curves

We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are H(ζ) = U+ U^{- 1}+ V+ ζ^{V - 1} and _{Hm, n}= U+ V+ q^{- m n U - m V - n}, where U and V are self-adjoint Weyl operators satisfying UV= q²VU with q= e ^{i π b2} , b> 0 and ζ> 0 , m, n∈ N. We prove that H(ζ) and _{Hm, n} are self-adjoint operators with purely discrete spectrum on L²(R). Using the coherent state transform we find the asymptotical behaviour for the Riesz mean _{∑ j≥ 1 (λ- λj) +} as λ→ ∞ and prove the Weyl law for the eigenvalue counting function N(λ) for these operators, which imply that their inverses are of trace class.