The Legendre equation, $$\frac{d}{dx}((1-x^2)y') + \lambda y = 0$$ is an irregular Sturm-Liouville problem with $p(x) = 1 - x^2$, so that -1 and 1 are its singular points. I have been trying since yesterday to prove that its eigenvalues are necessarily integers without any kind of success. How can one prove it?
2026-04-21 21:23:21.1776806601
Prove that the eigenvalues of the Legendre equation are integers
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