Asymptotic equipartition of energy for differential equations in Hilbert space

Of concern are second order differential equations of the form $(d/dt - i{f_1}(A))(d/dt - i{f_2}(A))u = 0$. Here A is a selfadjoint operator and ${f_1},{f_2}$ are real-valued Borel functions on the spectrum of A. The Cauchy problem for this equation is governed by a certain one parameter group of unitary operators. This group allows one to define the energy of a solution; this energy depends on the initial data but not on the time t. The energy is broken into two parts, kinetic energy $K(t)$ and potential energy $P(t)$, and conditions on A, ${f_1},{f_2}$ are given to insure asymptotic equipartition of energy: ${\lim _{t \to \pm \infty }}K(t) = {\lim _{t \to \pm \infty }}P(t)$ for all choices of initial data. These results generalize the corresponding results of Goldstein for the abstract wave equation ${d^2}u/d{t^2} + {A^2}u = 0$. (In this case, ${f_1}(\lambda ) \equiv \lambda ,{f_2}(\lambda ) \equiv - \lambda$.)