The equation I'm interested in is(adsbygoogle = window.adsbygoogle || []).push({});

(1-u^{2})P''(u) - 2uP'(u) + (λ - βu - m^{2}/(1-u^{2}))P(u) = 0, where m takes on integer values, λ is an eigenvalue, and β is a known constant. Also, there is a change of basis where u = cosθ.

It is similar to the Legendre's associated equation

(1-u^{2})P''(u) - 2uP'(u) + (λ - m^{2}/(1-u^{2})P(u) = 0, but includes the βu.

In the Legendre's associated equation, λ is constrained by eigenvalues of the forml(l+1), whereltakes on positive integer values.

I'll be looking around for solutions to the equation in question, but I would at least like confirmation as to whether λ here is constrained to integral values (in other words, the nature of the eigenspectrum). I suspect that they are not.

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# Sturm-Liouville System

Can you offer guidance or do you also need help?

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