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Eigenvalue Analysis 175
It is well-known that (8.6.1) has a countable set of positive eigenvalues λm = π2
4 m2 ,
m ≥ 1, corresponding to the normalized eigenfunctions: φm(x) = sin π
2
m even, φm(x) = cos π
2
mx, x ∈ Ī,
mx, x ∈ Ī, m odd. Besides, for any n ≥ 2, we have the
eigenvalues λn,m, 1 ≤ m ≤ n −1, of problem (8.2.1). These will be assumed to be real,
distinct and in increasing order.
We are concerned with the asymptotic behavior of the λn,m’s when n tends to
infinity. Actually, we expect the following convergence result:
(8.6.2) ∀m ≥ 1, lim
n→+∞ λn,m = λm.
This would suggest a technique for approximating the eigenvalues of differential opera-
tors, which is a crucial problem in many applications of mathematical physics (see for
instance courant and hilbert(1953), Vol.1). This procedure is investigated for the
solution of a large number of problems in calogero and franco (1985). In that pa-
per, discretizations of the derivative operators (denoted by Lagrangian differentiations)
are obtained starting from an arbitrary set of nodes, not necessarily related to Gauss
type formulas. Heuristic estimates of the rate of convergence of the limit in (8.6.2) are
presented in calogero(1983) for equispaced nodes and durand(1985) for Gauss type
nodes. According to these papers the convergence is exponential, and, as pointed out by
the authors in their numerical experiments, the results are definitely better than those
obtained by other techniques, such as the finite-differences method.
Remarks about the behavior of the eigenvalues in the Chebyshev case can be found
in weideman and trefethen(1988). The authors make the following observation.
For any fixed n ≥ 2, about two thirds of the eigenvalues in (8.2.1) are very close to
the corresponding eigenvalues in (8.6.1). The remaining part of the spectrum deviates
sharply. This behavior justifies the estimate (8.3.6), since λn,n−1 turns out to be
proportional to n 4 , while λn just grows like n 2 . This fact is explained by the poor
approximation properties of the eigenfunctions with high frequency oscillations, where
the number of nodes is not sufficient to recover a satisfactory resolution (see section
6.8). This does not imply that (8.6.2) is false. In fact, for fixed m, the eigenvalue
λn,m begins to converge for sufficiently large n.