Of concern are second order differential equations of the form
(
d
/
d
t
−
i
f
1
(
A
)
)
(
d
/
d
t
−
i
f
2
(
A
)
)
u
=
0
(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
{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
)
K(t)
and potential energy
P
(
t
)
P(t)
, and conditions on A,
f
1
,
f
2
{f_1},{f_2}
are given to insure asymptotic equipartition of energy:
lim
t
→
±
∞
K
(
t
)
=
lim
t
→
±
∞
P
(
t
)
{\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
{d^2}u/d{t^2} + {A^2}u = 0
. (In this case,
f
1
(
λ
)
≡
λ
,
f
2
(
λ
)
≡
−
λ
{f_1}(\lambda ) \equiv \lambda ,{f_2}(\lambda ) \equiv - \lambda
.)