Let
L
=
∑
m
=
0
n
A
m
(
x
)
D
m
L = \sum \nolimits _{m = 0}^n {{A_m}(x){D^m}}
be a differential operator on
⊕
k
L
2
[
0
,
1
]
{ \oplus ^k}{L^2}[0,1]
with infinitely differentiable
k
×
k
k \times k
matrix valued coefficients. Assume that
det
A
n
(
x
)
≠
0
\det \,{A_n}(x) \ne 0
for
x
∈
[
0
,
1
]
x \in [0,1]
. The domain of L is the set of k-vector valued functions f such that
f
∈
C
n
−
1
(
[
0
,
1
]
)
,
f
(
n
−
1
)
f \in {C^{n - 1}}([0,1]),{f^{(n - 1)}}
is absolutely continuous on [0, 1] and
L
f
∈
⊕
k
L
2
[
0
,
1
]
Lf \in { \oplus ^k}{L^2}[0,1]
. Let
x
0
∈
(
0
,
1
)
{x_0} \in (0,1)
. Then there is a neighborhood
U
(
x
0
)
U({x_0})
containing
x
0
{x_0}
such that the restrictions of the eigenfunctions of L to
U
(
x
0
)
U({x_0})
have dense span in
⊕
k
L
2
[
U
(
x
0
)
]
{ \oplus ^k}{L^2}[U({x_0})]
. The example
L
=
e
−
i
x
d
/
d
x
L = {e^{ - ix}}d/dx
shows that this is the best possible abstract result.