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Sturm-Liouville Problem

How could one prove that there are at most countably many eigenvalues of the Sturm-Liouville problem $−Lu=ju$, $j$ = eigenvalue, and $u$ is in $C^2[a,b]$?

I have put some more thought and I am still confused about this problem. Suppose I have $-Lu=\lambda u$ where $-Lu=(pu')'+qu.$ How can you justify that if $\left\{ c_{\alpha}\right\} _{\alpha\in\Gamma}$ is a set of non-zero real numbers such that $\mathbb{\sum}_{j=1}^{N}c_{\alpha_{j}}^{2}\leq1$ for every choice of $N\geq1$ and every choice of distinct $\alpha.$ Then $\Gamma$ is a countable set. How to proceed from there? Is there an easy solution? Thank you very much for help.

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    I'm familiar with the line of proof that Wikipedia mentions, which is: show that the Sturm-Liouville operator is compact and apply the spectral theorem for compact operators (which is considerably easier to prove than the general one). I take it that you are looking for a more direct way?2011-05-20

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