m-functions and Floquet exponents for linear differential systems
type: Item
instance of: scholarly article
title: m-functions and Floquet exponents for linear differential systems
zbMATH Open document ID: 0652.34016
DOI: 10.1007/BF01762419
author: Russell Johnson
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1987-01-01T00:00:00Z
review text: Let J be a nonsingular k by k matrix with J \(*=-J\). We can assume that \(J=-iI_ p\oplus iI_ q\) \((p+q=k)\), \(I_ r\) denoting the r-dimensional identity matrix. Consider a linear system of ordinary differential equations \((2)_{y,\lambda}\) \(dx/dt=(\lambda J^{-1}\gamma_ y(t)+y(t))x\) where \(x\in {\mathbb{C}}^ k,\) \(\lambda\in {\mathbb{C}}\), \(\gamma_ y(t)\) is a k by k matrix valued function of t depending on y(\(\cdot)\) with \(\gamma\) \(*_ y=\gamma_ y\), \(\gamma_ y\geq 0\). The y(t) is a k by k matrix valued function of t which is stationary ergodic and satisfies a symmetry condition, that is, for every \(t\in {\mathbb{R}}\), y(t) belongs to the Lie algebra of the linear Lie group which preserves \(<x,Jy>\) where \(<,>\) is the Euclidean inner product on \({\mathbb{C}}^ k.\) The author first defines a function w(\(\lambda)\) for real \(\lambda\) which is shown to have a meaning like Floquet exponents dfined for periodic linear differential equations. Next he obtains Weyl-Kodaira m-functions \(m_+(\lambda)\), \(m_-(\lambda)\) for \((2)_{y,\lambda}\) by using the fact that \((2)_{y,\lambda}\) has exponential dichotomy for Im \(\lambda\neq 0\). Then he defines by means of these m-functions a function w(\(\lambda)\) holomorphic for Im \(\lambda\) \(>0\) (or \(<0)\). It is shown that the boundary values of w(\(\lambda)\) on the real axis coincides to those of w(\(\lambda)\) defined first. Lastly, the author applies the function w(\(\lambda)\) to the spectral theory of \((2)_{y,\lambda}\).
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34M99
Mathematics Subject Classification ID: 34L99
Mathematics Subject Classification ID: 35P99
zbMATH DE Number: 4063535
zbMATH Keywords: Lie algebra
zbMATH Keywords: Floquet exponents
zbMATH Keywords: Weyl-Kodaira m-functions
reviewed by: Kyoichi Takano
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01762419
OpenAlex ID: W2061444724
description: scientific article; zbMATH DE number 4063535