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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
Article

Abstract

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.

Keywords

Coherent State Topological String Trace Class Tauberian Theorem Dehn Twist 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© 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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