JOURNAL OF DIFFERENTIAL EQUATIONS 89, 110-120 (1991) Asymptotic Partition of Energy for Abstract Uniformly Propagative Systems RAINER H. PICARD Department of Mathematical Sciences, University of Wisconsin, Milwuukee, Milwaukee, Wisconsin 5321 I Received August 15, 1989 0. INTRODUCTION The last three decades have seen a considerable activity in the subject of partition of energy for conservative systems. The topic going back to a first mathematically rigorous result in Lax and Phillips’ book [4] has quickly developed into a subject of functional analytical interest. The central issue is familiar from the analysis of various conservative physical systems which display the property that kinetic energy tends to equate potential energy (equi-partition of energy). The mechanism providing this feature has been analyzed extensively. A crucial observation in the problem’s resolution that can be traced back to Goldstein and Sandefur [2] is that the property of equi-partition of energy is due to the particular structure of certain operator matrices. The partition results then take on the form of equal partition of the contributions of the components to the complete (energy) norm of the solution of an associated evolution equation. Two different main paths have been taken to lead the topic closer to a satisfactory con- clusion, (compare, however, the results of [S] for a different generalization of independent interest). One path is limited to particular 2 x 2 operator matrices composed of not necessarily commuting operators which, however, covers a wide range of physical phenomena [7]. The size limita- tion is basically due to the intention of avoiding unmotivated and awkward commutator relationships. Another path of investigation can be developed by pursuing the question of partition of energy for larger operator matrices by making the sacrifice of assuming that all operators constituting the matrix commute, compare [2,6]. The considerations in [6] seem to give a fairly far-reaching answer to the problem in question. The involvement of discrete Fourier transforms in this context, however, seemed to be mysterious and an explanation of this fact could not be provided. The pre- sent considerations will address precisely this point. It turns out that based on an idea that has been developed by D. Goldstein-Costa [3] to obtain 110 0022~0396/91 $3.00 Copyright 0 1991 by Academtc Press, Inc All rights of reproduction in any form reserved