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

• Published in: Geometric and functional analysis Vol. 26; no. 1; pp. 288 - 305
• Main Authors: Laptev, Ari, Schimmer, Lukas, Takhtajan, Leon A.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.02.2016
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Tauberian Theorem; Trace Class; Dehn Twist; Topological String; Coherent State
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-016-0357-8

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 H m , n = U + V + q - m n U - m V - n , where U and V are self-adjoint Weyl operators satisfying U V = q 2 V U with q = e i π b 2 , b > 0 and ζ > 0 , m , n ∈ N . We prove that H ( ζ ) and H m , n are self-adjoint operators with purely discrete spectrum on L 2 ( R ) . Using the coherent state transform we find the asymptotical behaviour for the Riesz mean ∑ j ≥ 1 ( λ - λ j ) + 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.