Poles of integrále tritronquée and anharmonic oscillators. A WKB approach

Masoero, Davide

doi:10.1088/1751-8113/43/9/095201

Cited by 42 publications

(110 citation statements)

References 23 publications

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“…The precision degrades quickly for the poles further from the origin, but this can be improved by taking larger N , and also by combining with trans-asymptotic estimates such as (31), which become much more precise for the poles further from the origin. In this paper we have not implemented these refinements, but we quote these initial values because of their relevance to the quantum mechanical spectral problem for cubic oscillators, and because only very low order values exist in the literature for the first two real poles, x 1 and x 6 [58]. A more detailed numerical study of the tritronquée pole values is left for future work.…”

Section: Fine Structure Of the Tritronquée Polesmentioning

confidence: 99%

“…A more detailed numerical study of the tritronquée pole values is left for future work. This is motivated by results connecting poles of PI solutions to spectral properties of cubic oscillators [58][59][60], in analogy to results relating pole behavior of Painlevé III solutions with the spectrum of the Mathieu equation [61][62][63], and pole behavior of Painlevé VI solutions to spectral properties of an associated Heun equation [64,65].…”

Section: Fine Structure Of the Tritronquée Polesmentioning

confidence: 99%

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Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I

Costin

Dunne

2019

J. Phys. A: Math. Theor.

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Extrapolation is a generic problem in physics and mathematics: how to use asymptotic data in one parametric regime to learn about the behavior of a function in another parametric regime. For example: extending weak coupling expansions to strong coupling, or high temperature expansions to low temperature, or vice versa. Such extrapolations are particularly interesting in systems possessing dualities. Here we study numerical procedures for performing such an extrapolation, combining ideas from resurgent asymptotics with well-known techniques of Borel summation, Padé approximants and conformal mapping. We illustrate the method with the concrete example of the Painlevé I equation, which has applications in many branches of physics and mathematics. Starting solely with a finite number of coefficients from asymptotic data at infinity on the positive real line, we obtain a high precision extrapolation of the function throughout the complex plane, even across the phase transition into the pole region. The precision far exceeds that of state-of-the-art numerical integration methods along the real axis. The methods used are both elementary and general, not relying on Painlevé integrability properties, and so are applicable to a wide class of extrapolation problems.

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Section: Fine Structure Of the Tritronquée Polesmentioning

confidence: 99%

Section: Fine Structure Of the Tritronquée Polesmentioning

confidence: 99%

Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I

Costin

Dunne

2019

J. Phys. A: Math. Theor.

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“…It was proved recently in [13] with a technique developed in [12]; see also [25][26][27] for partial results.…”

Section: Conjecture 11 If the 2-or 3-truncated Solution Of A Painlevmentioning

confidence: 99%

“…The locations of the movable poles for the Painlevé transcendents are crucial for understanding a number of problems arising from mathematical physics; cf. [4,16,26,27]. In the pioneering works [6,7], Boutroux established the "deformed" elliptic function approximations in appropriate sectors near infinity, which leads to the degeneration of lattices of poles along the critical rays This means that the poles tend to align themselves along certain smooth curves which tend to one of the rays Γ k near infinity.…”

Section: Introductionmentioning

confidence: 99%

Location of Poles for the Hastings–McLeod Solution to the Second Painlevé Equation

Huang

Zhang

2015

Constr Approx

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We show that the well-known Hastings-McLeod solution to the second Painlevé equation is pole-free in the region arg x ∈ [− π 3 , π 3 ] ∪ [ 2π 3 , 4π 3 ], which proves an important special case of a general conjecture concerning pole distributions of Painlevé transcedents proposed by Novokshenov. Our strategy is to construct explicit quasi-solutions approximating the Hastings-McLeod solution in different regions of the complex plane and estimate the errors rigorously. The main idea is very similar to the one used to prove Dubrovin's conjecture for the first Painlevé equation, but there are various technical improvements.

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“…A similar interpretation for PVI is consistent with the quantum geometry and exact WKB explanations of the P/NP relations for all genus 1 systems, in terms of all-orders actions and dual actions[55,57,70,[72][73][74][75][76][77][78][79].3. A similar, but not identical, relation between spectral problems and Painlevé equations arisesfor the PI equation, whose tritronquée poles have been associated with a leading order WKB analysis of the cubic QM oscillator[69,80,81].4. Another identification of Painlevé equations with spectral boundary value problems arises in the study of PT symmetric quantum mechanics[82].…”

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confidence: 99%

Resurgence, Painlevé equations and conformal blocks

Dunne

2019

J. Phys. A: Math. Theor.

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We discuss some physical consequences of the resurgent structure of Painlevé equations and their related conformal block expansions. The resurgent structure of Painlevé equations is particularly transparent when expressed in terms of physical conformal block expansions of the associated tau functions. Resurgence produces an intricate network of inter-relations; some between expansions around different critical points, others between expansions around different instanton sectors of the expansions about the same critical point, and others between different non-perturbative sectors of associated spectral problems, via the Bethe-gauge and Painlevé-gauge correspondences. Resurgence relations exist both for convergent and divergent expansions, and can be interpreted in terms of the physics of phase transitions. These general features are illustrated with three physical examples: correlators of the 2d Ising model, the partition function of the Gross-Witten-Wadia matrix model, and the full counting statistics of one dimensional fermions, associated with Painlevé VI, Painlevé III and Painlevé V, respectively.

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Poles of integrále tritronquée and anharmonic oscillators. A WKB approach

Cited by 42 publications

References 23 publications

Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I

Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I

Location of Poles for the Hastings–McLeod Solution to the Second Painlevé Equation

Resurgence, Painlevé equations and conformal blocks

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