Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point, I — Its WKB-theoretic transformation to the Mathieu equation

Kamimoto, Shingo; Kawai, Takahiro; Takei, Yoshitsugu

doi:10.1016/j.aim.2014.02.026

Cited by 13 publications

(8 citation statements)

References 3 publications

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“…Later developments focused mainly on understanding WKB-theoretic transformation series (first introduced in [AKT91]) that transform a given differential equation in a suitable neighbourhood of critical WKB trajectories (or Stokes lines) to one in standard form whose WKB-theoretic properties are better understood. A very partial list of contributions includes the works by Aoki, Kamimoto, Kawai, Koike, Sasaki, and Takei, [AKT09, KK11, KKT12, KK13,Sas13,KKT14a]. Parallel to this activity has been the classification of WKB geometry (or Stokes graphs), which includes the works by Aoki, Kawai, Takei, Tanda, [AKT01, Tak07, AT13, Tak17], as well as a detailed analysis of some WKB-theoretic properties of special classes of equations, which includes the works by Aoki, Kamimoto, Kawai, Koike, and Takei [KT11, KKKT11, ATT14, KKT14b, Tan15, KKK16, ATT16, AT16, AIT19].…”

Section: Resultsmentioning

confidence: 99%

Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs

Nikolaev¹

2021

Preprint

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We prove an existence and uniqueness theorem for exact WKB solutions of general singularly perturbed linear second-order ODEs in the complex domain. These include the one-dimensional time-independent complex Schrödinger equation. Notably, our results are valid both in the case of generic WKB trajectories as well as closed WKB trajectories. We also explain in what sense exact and formal WKB solutions form a basis. As a corollary of the proof, we establish the Borel summability of formal WKB solutions for a large class of problems, and derive an explicit formula for the Borel transform.

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Section: Resultsmentioning

confidence: 99%

Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs

Nikolaev¹

2021

Preprint

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“…For more recent and advanced works related to WKB analysis and local/global studies of solutions to linear ODEs near turning points, we refer to contributions related to the 1D Schrödinger equation with simple poles [11], with merging pairs of simple poles and turning points [12], and with merging triplet of poles and turning points [13,14] and for analytic continuation properties of the Borel transform (resurgence) of WKB expansions in the problem of confluence of two simple turning points we quote [15]. Concerning the structure of singular formal solutions to singularly perturbed linear systems of ODEs with turning points we point out [16] solving an old question of Wasow.…”

Section: ̂(mentioning

confidence: 99%

On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity

Malek

2017

Abstract and Applied Analysis

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We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter . The problem involves an irregular singularity in time, as in a recent work of the author and A. Lastra, but possesses also, as a new feature, a turning point at the origin in C. We construct a family of sectorial meromorphic solutions obtained as a small perturbation in of a slow curve of the equation in some time scale. We show that the nonsingular parts of these solutions share common formal power series (that generally diverge) in as Gevrey asymptotic expansion of some order depending on data arising both from the turning point and from the irregular singular point of the main problem.

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“…Keeping these previous works in mind, to investigate the relation between BPS structures and TR, we study the Stokes structures of the Voros symbols for the quantum curves constructed through TR from the spectral curves of hypergeometric type obtained in [IKoT1,IKoT2], and compare those to the BPS Riemann-Hilbert problems associated with the BPS structure studied in our previous work [IK]. We note here that such analysis of the Stokes phenomenon for the Voros symbols of the (confluent) hypergeometric differential equations are not new; some special cases (e.g., for specific ν) have already been studied in the exact WKB literature (see [AT,ATT1,AIT,T,KoT,KKKoT,KKT] for example). We also note that Stokes structures of the Voros symbols for Schrödinger-type equations were also studied in [DDP1,IN1] in a general setting.…”

Section: Introductionmentioning

confidence: 63%

“…As we mentioned in §3.4, Γ * is generated by β s , s ∈ P ev , which is a class represented by a path from s − to s + . The path Voros coefficients of the hypergeometric differential equations for those paths were studied in[SS,T,KoT, AT,KKT,ATT1,AIT,IKoT1,IKoT2] etc. Let us recall and rewrite the formula for their explicit expression, following[IKoT2,IK].…”

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confidence: 99%

Topological recursion and uncoupled BPS structures II: Voros symbols and the $τ$-function

Iwaki¹,

Kidwai²

2021

Preprint

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We continue our study of the correspondence between BPS structures and topological recursion in the uncoupled case, this time from the viewpoint of quantum curves. For spectral curves of hypergeometric type, we show the Borel-resummed Voros symbols of the corresponding quantum curves solve Bridgeland's "BPS Riemann-Hilbert problem". In particular, they satisfy the required jump property in agreement with the generalized definition of BPS indices Ω in our previous work. Furthermore, we observe the Voros coefficients define a closed one-form on the parameter space, and show that (log of) Bridgeland's τ -function encoding the solution is none other than the corresponding potential, up to a constant. When the quantization parameter is set to a special value, this agrees with the Borel sum of the topological recursion partition function Z TR , up to a simple factor. KOHEI IWAKI AND OMAR KIDWAI 4.7. Stokes phenomenon for Voros symbols 35 4.8. The Voros potential 36 5. Solution to BPS Riemann-Hilbert problem and the τ -function 37 5.1. BPS indices and the Stokes structure of the Voros coefficients 37 5.2. Relation to the minimal and holomorphic solutions 39 5.3. BPS τ -function and the Voros potential 41 5.4. Borel-resummed TR partition function as a BPS τ -function 43 5.5. Higher degree spectral curves 44 Appendix A. Explicit expression for Voros coefficients 46 References 47

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Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point, I — Its WKB-theoretic transformation to the Mathieu equation

Cited by 13 publications

References 3 publications

Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs

Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs

On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity

Topological recursion and uncoupled BPS structures II: Voros symbols and the $τ$-function

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