Differential equation similar to Legendre

Main Question or Discussion Point

I am trying to solve the following differential equation:

$$(\frac{L^2}{6k^2}+\frac{w\sqrt{3}}{2}\sin^2\theta\cos 2\phi)\psi=E\psi$$

where $$L^2$$is the angular momentum given by:
$$\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}(sin\theta\frac{partial}{\partial\theta})-\frac{1]{sin^2\theta}\frac{\partial^2}{\partial\phi^2}$$. $$\theta$$ goes from 0 to $$\pi$$ while $$\phi$$ goes from 0 to 2 $$\pi$$. $$k$$ and $$w$$ are constants and E is the energy of the system.. This differential equation seems non separable. Any ideas how to solve it...I also realised that the term $$sin^2\theta\cos 2\phi$$ is a combination of $$(Y_{2,-2]+ Y_{2,2})$$. But then how to continue?

Thanks

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Sorry L^2 is:
$$\frac{1}{\sin\theta}\frac{\partial}{\partial\theta} (\sin\theta\frac{partial}{\partial\theta})-\frac{1]{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}$$

You mean

$$L^2 = \frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)-\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}.$$

Some ideas: apply the operator to $Y_L^M$ with arbitary values of $L,\,M$ and see if you can determine $L,\,M$ using the properties of the polynomials, or maybe using a linear combination of Legendre polynomials, or appliyng the operator to $f(\theta)Y_L^M,\,g(\phi)Y_L^M,\,h(\theta,\phi)Y_L^M$ and use the properties of the polynomials to simplify the equation and determine $f,\,g,\,h$.

Just ideas.

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