m-functions and Floquet exponents for linear differential systems
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 \((2)_{y,\lambda}\).
scientific article; zbMATH DE number 4063535
scientific article