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+ζV−1H(ζ)=U+U−1+V+ζV−1 and Hm,n=U+V+q−mnU−mV−nHm,n=U+V+q−mnU−mV−n, where UU and VV are self-adjoint Weyl operators satisfying UV=q2VUUV=q2VU with q=eiπb2q=eiπb2, b>0b>0 and ζ>0ζ>0, m,n∈Nm,n∈N. We prove that H(ζ)H(ζ) and Hm,nHm,n are self-adjoint operators with purely discrete spectrum on L2(R)L2(R). Using the coherent state transform we find the asymptotical behaviour for the Riesz mean ∑j≥1(λ−λj)+∑j≥1(λ−λj)+ as λ→∞λ→∞ and prove the Weyl law for the eigenvalue counting function N(λ)N(λ) for these operators, which imply that their inverses are of trace class.