Separated Dirac operators and asymptotically constant linear systems

Abstract

Levinson's theorem with all its ramifications is a well-established tool in the spectral analysis of ordinary differential operators (see in particular §§3·10, 11 and 4·3, 4 of Eastham's book [5]). In this note we should like to draw attention to a result that describes the solutions of asymptotically constant linear systems under weaker assumptions less precisely than the Levinson theorem. This result can be called the Perron–Lettenmeyer–Hartman–Wintner theorem after the contributions of these authors in [11, 10, 6, 8]. (Note that the Hartman–Wintner theorem that is discussed in [5, p. 17] is a substantially different version of this theorem.)