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Eigenvalue Analysis 155
We find from numerical tests that the λn,m’s have positive real part when −1 < β ≤ 0
and γ > γn, where γn, n ≥ 1, is a positive real constant. This result is found in
funaro and gottlieb(1988) for α = β = −1/2. Here, we give a simplified version in
the Legendre case.
Theorem 8.1.2 - Let α = β = 0. Then, for any n ≥ 1, there exists a constant
γn > 0, such that, for γ > γn, all the eigenvalues of problem (8.1.6) satisfy Reλn,m > 0,
0 ≤ m ≤ n.
Proof - We first note that pn,m(η (n)
0 ) = 0, 0 ≤ m ≤ n. Otherwise, we would have the
differential equation p ′ n,m = λn,mpn,m which is satisfied only for pn,m ≡ 0. Then, for
0 ≤ m ≤ n, we get the relations (the upper bar denotes the complex conjugate)
(8.1.7)
d
dx (|pn,m| 2 )(η (n)
i ) = (pn,m¯p ′ n,m + p ′ n,m¯pn,m)(η (n)
i )
= ( ¯ λn,mpn,m¯pn,m + λn,mpn,m¯pn,m)(η (n)
i ) = 2[|pn,m|(η (n)
i )] 2 Reλn,m, 1 ≤ i ≤ n,
(8.1.8) 2γ[|pn,m|(η (n)
0 )] 2 + d
dx (|pn,m| 2 )(η (n)
0 ) = 2[|pn,m|(η (n)
0 )] 2 Reλn,m.
We multiply the expressions in (8.1.7), (8.1.8) by the weights ˜w (n)
i , 0 ≤ i ≤ n, given in
(3.5.6). By summing over i, one obtains
(8.1.9) 2γ[|pn,m|(η (n)
0 )] 2 ˜w (n)
0 +
= 2Reλn,m
n
i=0
n
i=0
d
dx (|pn,m| 2 )(η (n)
i ) ˜w (n)
i
[|pn,m|(η (n)
i )] 2 ˜w (n)
i , 0 ≤ m ≤ n.
Since the weights are positive, it is sufficient to show that the term on the left-hand
side of (8.1.9) is also positive. By virtue of formula (3.5.1) and theorem 3.5.1 (w ≡ 1),
considering that d
dx (|pn,m| 2 ) ∈ P2n−1, we get