Frequently, nonconservative semigroups generated by partial differential operators in
L
2
,
ρ
(
R
k
)
{L_{2,\rho }}({R^k})
have the property that initial conditions which are large at
|
x
|
=
∞
|x| = \infty
become immediately small at infinity for all
t
>
0
t > 0
. This property is related to the rate of decay of eigenfunctions of the differential operator. In this paper this phenomenon is investigated for a large class of differential operators of second and higher order. New estimates on the rate of decay of the eigenfunctions are included, which are related in special cases to those of Agmon.