Untitled - Cdm.unimo.it

160 Polynomial Approximation of Differential Equations
Theorem 8.2.3 - Let α = β = 0. Then, for any n ≥ 2, the eigenvalues of problem
(8.2.1) are real and strictly positive.
Proof - Given that w ≡ 1, it is sufficient to recall the equality
(8.2.11)
1
p
−1
′ n,m(¯pn,mw) ′ dx =
1
|p
−1
′ n,m| 2 dx > 0, 1 ≤ m ≤ n − 1.
For different values of the parameters α and β, problem (8.2.1) still furnishes real
eigenvalues, though the proof of this statement is more involved than for the Legendre
case. By studying the roots of the characteristic polynomial, gottlieb and lust-
man(1983) give the proof in the Chebyshev case. To this end, they analyze the eigen-
values of two auxiliary first-order problems. Other cases are still an open question. For
−1 < α < 1 and −1 < β < 1 , the numerical experiments generate real eigenvalues,
and some complex eigenvalues with relatively small imaginary part. In general, the
analysis of the eigenvalue problem related to (7.4.12) does not lead to real eigenvalues,
due to the presence of the first-order derivative (see previous section). However, when
A and B are sufficiently close to zero, the second order operator dominates, and the
eigenvalues are a small perturbation of those in (8.2.1). Thus, their real part is positive.
We examine the eigenvalue problems associated to (7.4.13) and (7.4.16). Namely,
for 0 ≤ m ≤ n, we have
⎧
(8.2.12)
(8.2.13)
⎪⎨
⎪⎩
⎧
⎪⎨
⎪⎩
−p ′′ n,m(η (n)
i ) + µpn,m(η (n)
i ) = λn,mpn,m(η (n)
i ), 1 ≤ i ≤ n − 1,
−p ′ n,m(η (n)
0 ) = λn,mpn,m(η (n)
0 ),
p ′ n,m(η (n)
n ) = λn,mpn,m(η (n)
n ),
−p ′′ n,m(η (n)
i ) + µpn,m(η (n)
i ) = λn,mpn,m(η (n)
i ), 1 ≤ i ≤ n − 1,
−p ′′ n,m(η (n)
0 ) + µpn,m(η (n)
0 ) − γp ′ n,m(η (n)
0 ) = λn,mpn,m(η (n)
0 ),
−p ′′ n,m(η (n)
n ) + µpn,m(η (n)
n ) + γp ′ n,m(η (n)
n ) = λn,mpn,m(η (n)
n ),
where µ > 0 and γ ∈ R with γ = 0.