On the exact WKB analysis of higher order simple-pole type operators
type: Item
instance of: scholarly article
title: On the exact WKB analysis of higher order simple-pole type operators
zbMATH Open document ID: 1231.34155
author: Takahiro Kawai
author: Tatsuya Koike
author: Yoshitsugu Takei
published in: Advances in Mathematics
publication date: 2011-08-19T00:00:00Z
review text: The authors recently introduced a class of higher-order simple-pole type linear differential operators with a large parameter \(\eta\) \[ P(x,d_x,\eta)=d_x^m+(\eta A_{1,0}(x)+x^{-1}A_{1}(x,\eta))\,d_x^{m-1}+x^{-1}\sum_{j=0}^{m-2}A_{j}(x,\eta)\,d_x^j, \] where \(A_{j}(x,\eta)\) are formal series in \(\eta^{-1}\) with coefficients holomorphic in \(x\). In the present paper, they prove a decomposition theorem for this operator \(P\), namely, \(xP=Q^{(1)}\cdots Q^{(m-2)}xR\), where \(Q^{(l)}=d_x-\eta q^{(l)}(x,\eta)\) are first order operators and \(R=R(x,d_x,\eta)\) is a second order simple-pole type differential operator which is uniquely determined by \(P(x,d_x,\eta)\) regardless the choice of \(Q^{(l)}\). Thus, from the point of view of exact WKB analysis, the Stokes geometry of any higher-order simple-pole type operator reduces in essence to the study of a Schrödinger operator.
reviewed by: Andrei A. Kapaev
Mathematics Subject Classification ID: 34M60
Mathematics Subject Classification ID: 47B40
zbMATH DE Number: 5941206
zbMATH Keywords: exact WKB analysis
zbMATH Keywords: simple-pole type operators
MaRDI profile type: Publication
full work available at URL: j.aim.2011.05.010
OpenAlex ID: W2064390734
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DOI: 10.1016/J.AIM.2011.05.010
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description: scientific article; zbMATH DE number 5941206