Differential equation similar to Legendre

[Physicslad78]

I am trying to solve the following differential equation:

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

where L^2is 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}<br />. \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 realized 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

 

[Physicslad78]

Sorry L^2 is:
<br /> \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}<br /> <br />

 

[AiRAVATA]

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.