Globally analytic simplification and the Levinson theorem
Consider the linear system of differential equations (*) \(dx/dt= A(t)x\) where the \(n\times n\)-matrix \(A(t)\) is represented in the form \(A(t)= \Lambda(t) +R(t)\) with \(\Lambda(t) =\text{diag} (\lambda_1(t), \dots, \lambda_n (t))\). The problem under consideration is to find conditions under which there exists a transformation \(x=[I+Q(t)]y\) reducing (*) to the diagonal system (**) \(dy/dt= \Lambda(t)y\). Corresponding results have been obtained, for example, by \textit{N. Levinson} [Duke Math. J. 15, 111-126 (1948; Zbl 0040.19402)] and \textit{W. A. Harris jun.} and \textit{Y. Sibuya} [Lect. Notes Math. 1475, 210-217 (1991; Zbl 0736.34010)]. In this paper, the authors apply results of global analytic simplification (diagonalization and triangularization) of a matrix function by the first two authors [Linear Algebra Appl. 169, 75-101 (1992; Zbl 0756.65033)] to study a class of systems (*) which can be reduced to one with analytic coefficients and such that \(Q(t)\) is globally analytic. Finally, the obtained results are illustrated by some examples.
scientific article; zbMATH DE number 567015