Gevrey Asymptotic Implicit Function Theorem

Nikita Nikolaev

doi:10.4171/LEM/1061

Gevrey Asymptotic Implicit Function Theorem

Nikita Nikolaev^*

^*Corresponding author for this work

Mathematics

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Abstract

We prove an Asymptotic Implicit Function Theorem in the setting of Gevrey asymptotics with respect to a parameter. The unique implicitly defined solution admits a Gevrey asymptotic expansion and furthermore it is the Borel resummation of the corresponding implicitly defined formal power series solution. The main theorem can therefore be rephrased as an Implicit Function Theorem for Borel summable power series. As an application, we give a diagonal or Jordan decomposition for holomorphic matrices in Gevrey asymptotic families.

Original language	English
Journal	L’Enseignement Mathématique
Early online date	11 Jan 2024
DOIs	https://doi.org/10.4171/LEM/1061
Publication status	E-pub ahead of print - 11 Jan 2024

Keywords

exact perturbation theory
singular perturbation theory
Borel summation
Borel-Laplace theory
asymptotic analysis
Gevrey asymptotics
resurgence
exact WKB analysis
complex analysis
Ordinary differential equations
WKB

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10.4171/LEM/1061

nikolaevn2022gevreyAccepted author manuscript, 447 KBLicence: None: All rights reserved

https://arxiv.org/abs/2112.08792Licence: None: All rights reserved

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 = "Gevrey Asymptotic Implicit Function Theorem",

abstract = "We prove an Asymptotic Implicit Function Theorem in the setting of Gevrey asymptotics with respect to a parameter. The unique implicitly defined solution admits a Gevrey asymptotic expansion and furthermore it is the Borel resummation of the corresponding implicitly defined formal power series solution. The main theorem can therefore be rephrased as an Implicit Function Theorem for Borel summable power series. As an application, we give a diagonal or Jordan decomposition for holomorphic matrices in Gevrey asymptotic families.",

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N2 - We prove an Asymptotic Implicit Function Theorem in the setting of Gevrey asymptotics with respect to a parameter. The unique implicitly defined solution admits a Gevrey asymptotic expansion and furthermore it is the Borel resummation of the corresponding implicitly defined formal power series solution. The main theorem can therefore be rephrased as an Implicit Function Theorem for Borel summable power series. As an application, we give a diagonal or Jordan decomposition for holomorphic matrices in Gevrey asymptotic families.

AB - We prove an Asymptotic Implicit Function Theorem in the setting of Gevrey asymptotics with respect to a parameter. The unique implicitly defined solution admits a Gevrey asymptotic expansion and furthermore it is the Borel resummation of the corresponding implicitly defined formal power series solution. The main theorem can therefore be rephrased as an Implicit Function Theorem for Borel summable power series. As an application, we give a diagonal or Jordan decomposition for holomorphic matrices in Gevrey asymptotic families.

KW - exact perturbation theory

KW - singular perturbation theory

KW - Borel summation

KW - Borel-Laplace theory

KW - asymptotic analysis

KW - Gevrey asymptotics

KW - resurgence

KW - exact WKB analysis

KW - complex analysis

KW - Ordinary differential equations

KW - WKB

U2 - 10.4171/LEM/1061

DO - 10.4171/LEM/1061

M3 - Article

SN - 0013-8584

JO - L’Enseignement Mathématique

JF - L’Enseignement Mathématique

ER -

Gevrey Asymptotic Implicit Function Theorem

Abstract

Keywords

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AbQuantumSpec - Abelianisation of Connections, Quantum Curves, and Spectral Clusters

Cite this