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I am trying to find Rayleigh quotient for this equation:

$u''(r) + [\frac{1-4n^2}{4r^2} + \lambda - 2n\beta -\beta^2r^2]u(r) = 0$, where $0 \le r \le 1$.

Is there any way to compute eigenvalue $\lambda$ by using Rayleigh quotient?

(The equation above is the normal form of $R''(r) + \frac{1}{r} R'(r) + [\lambda - (\frac{n}{r} - \beta r )^2]R(r) = 0$. )

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