Shatalov-Sternin's construction of complex WKB solutions and the choice of integration paths
type: Item
instance of: scholarly article
title: Shatalov-Sternin's construction of complex WKB solutions and the choice of integration paths
zbMATH Open document ID: 1312.34137
author: Alexander Getmanenko
published in: Mathematische Nachrichten
publication date: 2014-11-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1111.0834
Mathematics Subject Classification ID: 34M60
Mathematics Subject Classification ID: 34M03
zbMATH DE Number: 6368934
zbMATH Keywords: complex WKB
zbMATH Keywords: resurgence
MaRDI profile type: Publication
OpenAlex ID: W2963176549
arXiv ID: 1111.0834
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DOI: 10.1002/MANA.201100312
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review text: The author discusses methods of resurgent analysis to study the asymptotics of solutions of linear ODE with a small parameter. An equation of the type NEWLINE\[NEWLINE\left( 1 \right)\quad -{{\varepsilon }^{2}}\partial _{x}^{2}\varphi (\varepsilon ,\text{ }x)\text{ }+\text{ }V(x)\varphi (\varepsilon ,\text{ }x)=0,\quad NEWLINE\]NEWLINE where \(\varepsilon \) is a small complex parameter, and \(V(x)\) is an entire function, can be transformed by the Laplace transformation \(\int_{\gamma }{\Phi (s){{e}^{-{s}/{\varepsilon }\;}}ds}\) into an equation NEWLINE\[NEWLINE\partial _{x}^{2}\Phi (s,\text{ }x)\text{ }+\text{ }\partial _{s}^{2}V(x)\Phi (s,\text{ }x)=0,NEWLINE\]NEWLINE for an analytic function \(\Phi (s,\text{{ }}x)\) with a discrete set of singularities. It turns out that the terms of the hyperasymptotic expansions for (1) can be recovered from studying the singularities of \(\Phi (s,\text{{ }}x)\) see \textit{O. Costin} and \textit{S. Garoufalidis} [Ann. Inst. Fourier 58, No. 3, 893--914 (2008; Zbl 1166.34055)]. The author re-examines Shatalov-Sternin's proof of the existence of resurgent solutions for (1), focusing the attention on the study of the structure of the Riemann surface of \(\Phi \).
reviewed by: Mykola Grygorenko
description: scientific article; zbMATH DE number 6368934