Exact solutions for the singularly perturbed Riccati equation and exact WKB analysis

Nikita Nikolaev

doi:10.1017/nmj.2022.38

Exact solutions for the singularly perturbed Riccati equation and exact WKB analysis

Nikita Nikolaev

Mathematics

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Abstract

The singularly perturbed Riccati equation is the first-order nonlinear ordinary differential equation ℏ∂xf=af2+bf+c in the complex domain where ℏ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as ℏ→0 in a half-plane. These exact solutions are constructed using the Borel–Laplace method; that is, they are Borel summations of the formal divergent ℏ -power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schrödinger equation with a rational potential.

Original language	English
Pages (from-to)	1-36
Journal	Nagoya Mathematical Journal
Early online date	8 Dec 2022
DOIs	https://doi.org/10.1017/nmj.2022.38
Publication status	E-pub ahead of print - 8 Dec 2022

Keywords

exact perturbation theory
singular perturbation theory
Borel summation
Borel-Laplace theory
asymptotic analysis
Gevrey asymptotics
resurgence
exact WKB analysis
exact WKB method
nonlinear ODEs
Riccati equation
Schrödinger equation

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10.1017/nmj.2022.38Licence: Creative Commons: Attribution (CC BY)

NikolaevN2022ExactFinal published version, 867 KBLicence: Creative Commons: Attribution (CC BY)

https://arxiv.org/abs/2008.06492

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 = "Exact solutions for the singularly perturbed Riccati equation and exact WKB analysis",

abstract = "The singularly perturbed Riccati equation is the first-order nonlinear ordinary differential equation ℏ∂xf=af2+bf+c in the complex domain where ℏ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as ℏ→0 in a half-plane. These exact solutions are constructed using the Borel–Laplace method; that is, they are Borel summations of the formal divergent ℏ -power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schr{\"o}dinger equation with a rational potential.",

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

author = "Nikita Nikolaev",

year = "2022",

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doi = "10.1017/nmj.2022.38",

language = "English",

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journal = "Nagoya Mathematical Journal",

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AU - Nikolaev, Nikita

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N2 - The singularly perturbed Riccati equation is the first-order nonlinear ordinary differential equation ℏ∂xf=af2+bf+c in the complex domain where ℏ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as ℏ→0 in a half-plane. These exact solutions are constructed using the Borel–Laplace method; that is, they are Borel summations of the formal divergent ℏ -power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schrödinger equation with a rational potential.

AB - The singularly perturbed Riccati equation is the first-order nonlinear ordinary differential equation ℏ∂xf=af2+bf+c in the complex domain where ℏ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as ℏ→0 in a half-plane. These exact solutions are constructed using the Borel–Laplace method; that is, they are Borel summations of the formal divergent ℏ -power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schrödinger equation with a rational potential.

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 - exact WKB method

KW - nonlinear ODEs

KW - Riccati equation

KW - Schrödinger equation

U2 - 10.1017/nmj.2022.38

DO - 10.1017/nmj.2022.38

M3 - Article

SN - 0027-7630

SP - 1

EP - 36

JO - Nagoya Mathematical Journal

JF - Nagoya Mathematical Journal

ER -

Exact solutions for the singularly perturbed Riccati equation and exact WKB analysis

Abstract

Keywords

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

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