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Eigenvalue Analysis 157
Finally, the analysis of the eigenvalue problems associated to (7.4.7) and (7.4.8)
leads to positive real parts when α > −1, −1 < β ≤ 0, γ > γn > 0, and when A is a
non negative function.
8.2 Eigenvalues of higher-order operators
Let us begin with the discussion of second-order problems in the Jacobi case. The
eigenvalue problem corresponding to (7.4.9) consists in finding n+1 complex polynomials
pn,m, 0 ≤ m ≤ n, of degree at most n (n ≥ 2), and n + 1 complex numbers λn,m,
0 ≤ m ≤ n, such that
⎧
(8.2.1)
⎪⎨
⎪⎩
−p ′′ n,m(η (n)
i ) = λn,mpn,m(η (n)
i ), 1 ≤ i ≤ n − 1,
pn,m(η (n)
0 ) = λn,mpn,m(η (n)
0 ),
pn,m(η (n)
n ) = λn,mpn,m(η (n)
n ),
0 ≤ m ≤ n.
We assume that pn,0(η (n)
0 ) = 0 and pn,n(η (n)
n ) = 0. This implies that λn,0 = λn,n = 1.
The other n − 1 eigenvalues are obtained from the reduced system (7.4.9) after setting
σ1 = σ2 = 0 (see for instance (7.4.11), corresponding to the case n = 3). The relative
eigenfunctions satisfy pn,m(η (n)
0 ) = pn,m(η (n)
n ) = 0, 1 ≤ m ≤ n − 1.
To construct the characteristic polynomial, we argue as in theorem 8.1.1. We briefly
outline the proof. This time, we start from the relation
(8.2.2) −p ′′ n,m = λn,mpn,m + p ′′ n,m(−1) ˜ l (n)
0 + p′′ n,m(1) ˜ l (n)
n , 1 ≤ m ≤ n − 1.
We differentiate both the terms of equality (8.2.2), until the left-hand side vanishes.
Evaluating the resulting expression at the points x = ±1, leads to a 2 × 2 linear
system in the unknowns p ′′ n,m(−1) and p ′′ n,m(1). The eigenfunction pn,m in (8.2.2)
is not uniquely determined, since it can differ by a multiplicative constant. Hence, the
determinant of the 2 × 2 system must vanish. Imposing this final condition yields the
characteristic polynomial.