Geometric and Functional Analysis

, Volume 26, Issue 1, pp 288–305 | Cite as

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

  • Ari Laptev
  • Lukas Schimmer
  • Leon A. Takhtajan


We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are \({H(\zeta) = U + U^{-1} + V + \zeta V^{-1}}\) and \({H_{m,n} = U + V + q^{-mn}U^{-m}V^{-n}}\), where \({U}\) and \({V}\) are self-adjoint Weyl operators satisfying \({UV = q^{2}VU}\) with \({q = {\rm e}^{{\rm i}\pi b^{2}}}\), \({b > 0}\) and \({\zeta > 0}\), \({m, n \in \mathbb{N}}\). We prove that \({H(\zeta)}\) and \({H_{m,n}}\) are self-adjoint operators with purely discrete spectrum on \({L^{2}(\mathbb{R})}\). Using the coherent state transform we find the asymptotical behaviour for the Riesz mean \({\sum_{j\ge 1}(\lambda - \lambda_{j})_{+}}\) as \({\lambda \to \infty}\) and prove the Weyl law for the eigenvalue counting function \({N(\lambda)}\) for these operators, which imply that their inverses are of trace class.


Coherent State Topological String Trace Class Tauberian Theorem Dehn Twist 
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© Springer International Publishing 2016

Authors and Affiliations

  • Ari Laptev
    • 1
    • 2
  • Lukas Schimmer
    • 3
  • Leon A. Takhtajan
    • 4
    • 5
  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Institut Mittag-LefflerDjursholmSweden
  3. 3.Department of PhysicsPrinceton UniversityPrincetonUSA
  4. 4.Department of MathematicsStony Brook UniversityStony BrookUSA
  5. 5.Euler Mathematical InstituteSaint PetersburgRussia

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