Untitled - Cdm.unimo.it

Eigenvalue Analysis 163
(8.2.19) (1 − x 2 ) r IV
n,m(x) = n(n 2 − 1)(n + 2)ζn,mun(x) + {lower degree terms},
where un = P (ν,ν)
n . These relations can be checked by equating the coefficients of the
monomials x n . With the same arguments used in section 3.8 (we recall for instance
(3.8.12) and (3.8.15)), it is easy to show that
(8.2.20)
=
1
−1
n
i=0
r IV
n,m(η (n)
i )¯rn,m(η (n)
i ) ˜w (n)
i
r IV
n,m¯rn,m w dx + (un 2 w,n − un 2 w) n(n 2 − 1)(n + 2)|ζn,m| 2 .
Having unw,n ≥ unw, ∀n ≥ 1 (see (2.2.10) and (3.8.3)), integration by parts and
formula (8.2.16) yield
(8.2.21) Re
n
i=0
1
= Re r
−1
′′
n,m(¯rn,mw) ′′
dx
r IV
n,m(η (n)
i )¯rn,m(η (n)
i ) ˜w (n)
i
≥ C
1
−1
1
≥ Re r
−1
IV
n,m¯rn,mw dx
|r ′′
n,m| 2 w dx > 0, 1 ≤ m ≤ n − 1.
We note that, since rn,m(±1) = r ′ n,m(±1) = 0, the last integral in (8.2.21) does not
vanish (otherwise we would have rn,m ≡ 0). Hence, by (8.2.17) we deduce Reλn,m > 0,
1 ≤ m ≤ n − 1.
When ν = 0 (thus w ≡ 1), the right-hand side in (8.2.20) turns out to be real, after
integrating by parts twice.
Real and positive eigenvalues are also observed for other values of the parameters
α and β, and also for different type of boundary conditions such as those in (7.4.23).
Hints for the determination of the characteristic polynomial are given in funaro and
heinrichs(1990).