Asymptotics and Borel Summability

Costin, Ovidiu

doi:10.1201/9781420070323

Cited by 153 publications

(261 citation statements)

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“…For the convenience of the readers, we give a rough definition of Borel resummation method. (See [Cos09] for details.) Definition 2.5 (e.g., [KT05,§1]).…”

Section: Proposition 24 ([In14 Proposition 28])mentioning

confidence: 99%

Exact WKB Analysis and Cluster Algebras II: Simple Poles, Orbifold Points, and Generalized Cluster Algebras

Iwaki

Наканиши

2015

Int Math Res Notices

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Abstract. This is a continuation of developing mutation theory in exact WKB analysis using the framework of cluster algebras. Here we study the Schrödinger equation on a compact Riemann surface with turning points of simple-pole type. We show that the orbifold triangulations by Felikson, Shapiro, and Tumarkin provide a natural framework of describing the mutation of Stokes graphs, where simple poles correspond to orbifold points. We then show that under the mutation of Stokes graphs around simple poles the Voros symbols mutate as the variables of generalized cluster algebras introduced by Chekhov and Shapiro.

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“…For the convenience of the readers, we give a rough definition of Borel resummation method. (See [Cos09] for details.) Definition 2.5 (e.g., [KT05,§1]).…”

Section: Proposition 24 ([In14 Proposition 28])mentioning

confidence: 99%

Exact WKB Analysis and Cluster Algebras II: Simple Poles, Orbifold Points, and Generalized Cluster Algebras

Iwaki

Наканиши

2015

Int Math Res Notices

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“…The Borel-Padé approach is in essence a simple recipe to transform the coefficients of a divergent expansion into a table of approximants, which approximate the Borel sum of a divergent series and are typically evaluated by numerical contour integration. Another advantage of this approach is that it relies on widely studied approaches: conditions for Borel summability are rather well understood [41][42][43][44], and the properties of Padé approximantsused at a crucial step in the algorithm-are also very well known as they have been studied in depth for decades [45]. Given a divergent expansion ZðgÞ ∼ P ∞ n¼0 z n g n , where z n are the expansion coefficients and g is the expansion parameter ("the coupling"), the algorithm to calculate the Borel sum of a divergent series can be summarized as follows:…”

Section: A Borel-padé Resummationmentioning

confidence: 99%

Fast summation of divergent series and resurgent transseries from Meijer- G approximants

2018

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We develop a resummation approach based on Meijer-G functions and apply it to approximate the Borel sum of divergent series and the Borel-Écalle sum of resurgent transseries in quantum mechanics and quantum field theory (QFT). The proposed method is shown to vastly outperform the conventional Borel-Padé and Borel-Padé-Écalle summation methods. The resulting Meijer-G approximants are easily parametrized by means of a hypergeometric ansatz and can be thought of as a generalization to arbitrary order of the Borel-hypergeometric method [Mera et al., Phys. Rev. Lett. 115, 143001 (2015)]. Here we demonstrate the accuracy of this technique in various examples from quantum mechanics and QFT, traditionally employed as benchmark models for resummation, such as zero-dimensional ϕ 4 theory; the quartic anharmonic oscillator; the calculation of critical exponents for the N-vector model; ϕ 4 with degenerate minima; self-interacting QFT in zero dimensions; and the summation of one-and two-instanton contributions in the quantum-mechanical double-well problem.

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“…One of the most intriguing aspects of resurgent asymptotics [1][2][3][4][5], as applied to quantum theories, is that the characteristic divergence of fluctuations about certain saddle points, such as the perturbative vacuum, may encode detailed information about the global nonperturbative structure of the system. Two distinct types of resurgent behavior have been identified in quantum spectral problems.…”

Section: Introductionmentioning

confidence: 99%

Quantum geometry of resurgent perturbative/nonperturbative relations

Başar

Dunne

Ünsal

2017

J. High Energ. Phys.

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For a wide variety of quantum potentials, including the textbook 'instanton' examples of the periodic cosine and symmetric double-well potentials, the perturbative data coming from fluctuations about the vacuum saddle encodes all non-perturbative data in all higher non-perturbative sectors. Here we unify these examples in geometric terms, arguing that the all-orders quantum action determines the all-orders quantum dual action for quantum spectral problems associated with a classical genus one elliptic curve. Furthermore, for a special class of genus one potentials this relation is particularly simple: this class includes the cubic oscillator, symmetric double-well, symmetric degenerate triple-well, and periodic cosine potential. These are related to the Chebyshev potentials, which are in turn related to certain N = 2 supersymmetric quantum field theories, to mirror maps for hypersurfaces in projective spaces, and also to topological c = 3 Landau-Ginzburg models and 'special geometry'. These systems inherit a natural modular structure corresponding to Ramanujan's theory of elliptic functions in alternative bases, which is especially important for the quantization. Insights from supersymmetric quantum field theory suggest similar structures for more complicated potentials, corresponding to higher genus. Our approach is very elementary, using basic classical geometry combined with all-orders WKB.

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Asymptotics and Borel Summability

Cited by 153 publications

References 0 publications

Exact WKB Analysis and Cluster Algebras II: Simple Poles, Orbifold Points, and Generalized Cluster Algebras

Exact WKB Analysis and Cluster Algebras II: Simple Poles, Orbifold Points, and Generalized Cluster Algebras

Fast summation of divergent series and resurgent transseries from Meijer- G approximants

Quantum geometry of resurgent perturbative/nonperturbative relations

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