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

Лаптев, Ари; Schimmer, Lukas; Takhtajan, Leon A.

doi:10.1007/s00039-016-0357-8

Cited by 30 publications

(38 citation statements)

References 19 publications

(34 reference statements)

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“…This was later proved in [33,61]. With ρ S of trace class, one can then consider its Fredholm determinant (see for example [62]) 11) and it is a standard result that Ξ S (κ, ) is an entire function of κ.…”

Section: The Quantum Mechanical Proposalmentioning

confidence: 96%

Resurgence matches quantization

Couso-Santamaría¹,

Mariño²,

Schiappa³

2017

J. Phys. A: Math. Theor.

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Abstract:The quest to find a nonperturbative formulation of topological string theory has recently seen two unrelated developments. On the one hand, via quantization of the mirror curve associated to a toric Calabi-Yau background, it has been possible to give a nonperturbative definition of the topological-string partition function. On the other hand, using techniques of resurgence and transseries, it has been possible to extend the string (asymptotic) perturbative expansion into a transseries involving nonperturbative instanton sectors. Within the specific example of the local P 2 toric Calabi-Yau threefold, the present work shows how the Borel-Padé-Ecalle resummation of this resurgent transseries, alongside occurrence of Stokes phenomenon, matches the string-theoretic partition function obtained via quantization of the mirror curve. This match is highly non-trivial, given the unrelated nature of both nonperturbative frameworks, signaling at the existence of a consistent underlying structure.

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Section: The Quantum Mechanical Proposalmentioning

confidence: 96%

Resurgence matches quantization

Couso-Santamaría¹,

Mariño²,

Schiappa³

2017

J. Phys. A: Math. Theor.

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“…It was conjectured in [20] and then proved in [45,46] that this operator has a discrete spectrum and its inverse ρ = O −1 is trace class. Although these operators are not of Schrödinger type, and they lead to difference equations instead of differential equations, one can still use the all-orders WKB approximation [47], as it has been done in [34,48,50,80].…”

Section: Refined Holomorphic Anomaly Trans-series and Large-order Bementioning

confidence: 99%

“…It has been proved rigorously [45,46] that O −1 F 0 is trace class and positive when m F 0 > 0. In this paper we will focus on the special case m F 0 = 1, in which the theory simplifies considerably.…”

Section: (545)mentioning

confidence: 99%

Non-perturbative Quantum Mechanics from Non-perturbative Strings

Codesido

Mariño

Schiappa

2018

Ann. Henri Poincaré

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This work develops a new method to calculate non-perturbative corrections in onedimensional Quantum Mechanics, based on trans-series solutions to the refined holomorphic anomaly equations of topological string theory. The method can be applied to traditional spectral problems governed by the Schrödinger equation, where it both reproduces and extends the results of well-established approaches, such as the exact WKB method. It can be also applied to spectral problems based on the quantization of mirror curves, where it leads to new results on the transseries structure of the spectrum. Various examples are discussed, including the modified Mathieu equation, the double-well potential, and the quantum mirror curves of local P 2 and local F 0 . In all these examples, it is verified in detail that the trans-series obtained with this new method correctly predict the large-order behavior of the corresponding perturbative sectors.

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“…has nice properties: it has been shown in many examples in [1,2] that it is a positive definite trace class self-adjoint operator on L 2 (R), at least for appropriate values of ξ k . As such, it admits a discrete set of eigenvalues (−κ n ) −1 and eigenfunctions ψ n (x).…”

Section: The Difference Equation and Its Dualmentioning

confidence: 99%

“…When the genus g Σ of the underlying curve is greater or equal to one, the operator may have a discrete spectrum. More precisely, it has been shown in many examples that the inverse operator of the quantized curve is a positive definite trace class self-adjoint operator acting on L 2 (R) [1,2], implying that the spectrum of the operator is discrete.…”

Section: Introductionmentioning

confidence: 99%

Quantized mirror curves and resummed WKB

Zakany

2019

J. High Energ. Phys.

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Based on previous insights, we present an ansatz to obtain quantization conditions and eigenfunctions for a family of difference equations which arise from quantized mirror curves in the context of local mirror symmetry of toric Calabi-Yau threefolds. It is a first principles construction, which yields closed expressions for the quantization conditions and the eigenfunctions when /2π ∈ Q. The key ingredient is the modular duality structure of the underlying quantum integrable system. We use our ansatz to write down explicit results in some examples, which are successfully checked against purely numerical results for both the spectrum and the eigenfunctions. Concerning the quantization conditions, we also provide evidence that, in the rational case, this method yields a resummation of conjectured quantization conditions involving enumerative invariants of the underlying toric Calabi-Yau threefold.

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

Cited by 30 publications

References 19 publications

Resurgence matches quantization

Resurgence matches quantization

Non-perturbative Quantum Mechanics from Non-perturbative Strings

Quantized mirror curves and resummed WKB

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