2015/09/30 by Laptev, Ari, Schimmer, Lukas, Takhtajan, Leon A. · 2 citations
#34K08 #39A70 #47A75 #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1510.00045
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-1 and Hm,n=U+V+q-mnU-mV-n, where U and V are self-adjoint Weyl operators satisfying UV=q2VU with q=e^iπb2, 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.