A spectral notion for dynamic equations on time scales

Abstract

For linear dynamic equations x Δ= A( t) x in R N on a time scale T (e.g. T= Z or T= R ) the so-called dichotomy spectrum is introduced in this paper. This new spectrum consists of at most N closed intervals of the real line. In the autonomous case with T= R these intervals reduce to the real parts of the eigenvalues of A. In any case the spectral intervals are associated with invariant vector bundles comprising solutions with a common exponential growth rate. The main result of this paper is a spectral theorem which describes all possible forms of the dichotomy spectrum.