The Schrödinger operator
−
Δ
+
V
-\Delta +V
, of a compact Riemannian manifold
M
M
, has pure point spectrum. Suppose that
V
0
V_0
is a smooth reference potential. Various criteria are given which guarantee the compactness of all
V
V
satisfying
spec
(
−
Δ
+
V
)
=
spec
(
−
Δ
+
V
0
)
\operatorname {spec}(-\Delta +V)=\operatorname {spec}(-\Delta +V_0)
. In particular, compactness is proved assuming an a priori bound on the
W
s
,
2
(
M
)
W_{s,2}(M)
norm of
V
V
, where
s
>
n
/
2
−
2
s>n/2-2
and
n
=
dim
M
n=\dim M
. This improves earlier work of Brüning. An example involving singular potentials suggests that the condition
s
>
n
/
2
−
2
s>n/2-2
is appropriate. Compactness is also proved for non–negative isospectral potentials in dimensions
n
≤
9
n\le 9
.