Existence and uniqueness of exact WKB solutions for second-order singularly perturbed linear ODEs

Nikita Nikolaev

doi:10.1007/s00220-022-04603-7

Existence and uniqueness of exact WKB solutions for second-order singularly perturbed linear ODEs

Nikita Nikolaev^*

^*Corresponding author for this work

Mathematics

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Abstract

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.

Original language	English
Pages (from-to)	463-517
Number of pages	55
Journal	Communications in Mathematical Physics
Volume	400
Issue number	1
Early online date	17 Jan 2023
DOIs	https://doi.org/10.1007/s00220-022-04603-7
Publication status	Published - May 2023

Keywords

exact WKB analysis
exact WKB method
linear ODEs
Schrödinger equation
singular perturbation theory
exact perturbation theory
Borel resummation
Borel-Laplace theory
asymptotic analysis
exponential asymptotics
Gevrey asymptotics
resurgence

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10.1007/s00220-022-04603-7Licence: Creative Commons: Attribution (CC BY)

NikolaevN2023ExistenceFinal published version, 999 KBLicence: Creative Commons: Attribution (CC BY)

AbQuantumSpec - Abelianisation of Connections, Quantum Curves, and Spectral Clusters
Mazzocco, M. (Principal Investigator) & Nikolaev, N. (Co-Investigator)
European Commission
1/09/22 → 31/08/24
Project: EU

Cite this

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title = "Existence and uniqueness of exact WKB solutions for second-order singularly perturbed linear ODEs",

abstract = "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 Schr{\"o}dinger 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.",

keywords = "exact WKB analysis, exact WKB method, linear ODEs, Schr{\"o}dinger equation, singular perturbation theory, exact perturbation theory, Borel resummation, Borel-Laplace theory, asymptotic analysis, exponential asymptotics, Gevrey asymptotics, resurgence",

author = "Nikita Nikolaev",

year = "2023",

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doi = "10.1007/s00220-022-04603-7",

language = "English",

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journal = "Communications in Mathematical Physics",

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T1 - Existence and uniqueness of exact WKB solutions for second-order singularly perturbed linear ODEs

AU - Nikolaev, Nikita

PY - 2023/5

Y1 - 2023/5

N2 - 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.

AB - 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.

KW - exact WKB analysis

KW - exact WKB method

KW - linear ODEs

KW - Schrödinger equation

KW - singular perturbation theory

KW - exact perturbation theory

KW - Borel resummation

KW - Borel-Laplace theory

KW - asymptotic analysis

KW - exponential asymptotics

KW - Gevrey asymptotics

KW - resurgence

UR - https://doi.org/10.48550/arXiv.2106.10248

U2 - 10.1007/s00220-022-04603-7

DO - 10.1007/s00220-022-04603-7

M3 - Article

SN - 0010-3616

VL - 400

SP - 463

EP - 517

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 1

ER -

Existence and uniqueness of exact WKB solutions for second-order singularly perturbed linear ODEs

Abstract

Keywords

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