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m-functions and Floquet exponents for linear differential systems. (English) Zbl 0652.34016
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 $$(2)_{y,\lambda}$$.
Reviewer: K.Takano

##### MSC:
 34A30 Linear ordinary differential equations and systems 34M99 Ordinary differential equations in the complex domain 34L99 Ordinary differential operators 35P99 Spectral theory and eigenvalue problems for partial differential equations
##### Keywords:
Lie algebra; Floquet exponents; Weyl-Kodaira m-functions
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