In this paper we are concerned with solutions of the equation
Δ
u
+
p
(
x
)
u
=
0
\Delta u\, + \,p(x)u\, = \,0
in an unbounded domain
Ω
\Omega
in
R
n
{R^n}
,
Ω
⊃
{
x
|
‖
x
‖
⩾
R
0
}
\Omega \, \supset \,\{ x|\,\,\left \| x \right \|\, \geqslant \,{R_0}\}
. The main result is a determination of conditions on the asymptotic behavior of
p
(
x
)
p(x)
sufficient to guarantee that no nontrivial
L
2
{L_2}
solution exists. Our results contain those of previous authors as special cases. The principal application is to the determination of upper bounds for positive eigenvalues of Schrödinger operators.