Equipartition of energy for higher-order hyperbolic equations

Abstract

Let A(0), A(1),...,A(2(N)-1) be commuting skew-adjoint operators on a Hilbert space [unk]. Then the equation Pi(j=0) (2(N)-1) (d/dt - A(j))v(t) = 0 (t real) admits equipartition of energy [in the sense that the jth partial energy E(j)(t) of any solution at time t satisfies lim(t-->+/-infinity)E(j)(t) = 2(-N).(total energy) for each of the 2(N) values of j] if and only if the closure B(jk) of A(j) - A(k) satisfies weak-operator-limit exp(tB(jk)) = 0 as t --> +/-infinity whenever j not equal k.