Unformatted text preview: Given an eigenvalue λ, the corresponding
eigenspace is
Wλ = {f ∈ V0 : f satisﬁes (4.31)}.
Nonzero elements of the eigenspaces are called eigenfunctions .
If the functions ρ(x), σ (x) and κ(x) are complicated, it may be impossible to
solve this eigenvalue problem explicitly, and one may need to employ numerical
methods to obtain approximate solutions. Nevertheless, it is reassuring to know
that the theory is quite parallel to the constant coeﬃcient case that we treated
in previous sections. The following theorem, due to the nineteenth century
mathematicians Sturm and Liouville, is proven in more advanced texts:4
Theorem. Suppose that ρ(x), σ (x) and κ(x) are smooth functions which are
positive on the interval [a, b]. Then all of the eigenvalues of L are negative real
numbers, and each eigenspace is onedimensional. Moreover, the eigenvalues
can be arranged in a sequence
0 > λ 1 > λ2 > · · · > λ n > · · · ,
with λn → −∞. Finally, every wellbehaved function can be rep...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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