This paper is concerned with a centered finite-difference approximation to to the nonselfadjoint Sturm-Liouville eigenvalue problem
\[
L
[
u
]
=
−
[
a
(
x
)
u
x
]
x
−
b
(
x
)
u
x
+
c
(
x
)
u
=
λ
u
,
0
>
x
>
1
,
u
(
0
)
=
u
(
1
)
=
0.
\begin {array}{*{20}{c}} {L[u] = - {{[a(x){u_x}]}_x} - b(x){u_x} + c(x)u = \lambda u,\quad 0 > x > 1,} \hfill \\ {u(0) = u(1) = 0.} \hfill \\ \end {array}
\]
It is shown that the eigenvectors
W
p
{W_p}
of the
M
×
M
M \times M
-matrix (
Δ
x
=
1
/
(
M
+
1
)
\Delta x = 1/(M + 1)
mesh size), which approximates L, are bounded in the maximum norm independent of M if they are normalized so that
|
W
p
|
2
=
1
|{W_p}{|_2} = 1
.