The eigenvalues of low absolute value are

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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 one-dimensional. Moreover, the eigenvalues can be arranged in a sequence 0 > λ 1 > λ2 > · · · > λ n > · · · , with λn → −∞. Finally, every well-behaved function can be rep...
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