# Generic Sturm-Liouville Problem

The equation $\ddot{y}+(\lambda(r)^2-L^2r^{-2})y=0$ seems that can be cast in th eform of a generic Sturm-Liouville problem as $-\ddot{y}+q(r)y=\lambda_0\lambda(r)^2$ with $q(r)=L^2r^{-2}$ It can also be considered as a kind of normal form representation of more general equations but is there a general classification of possible solutions for (a) the Initial Value problem and (b) for the Boundary Value problem? From what is evident by inspection, for constant $\lambda=\lambda_0$ the problem becomes like the one for the Schroedinger equation with a potential $V(r)=L^2r^{-2}$ in which case spherical Bessel functions should be sufficient. A search with the use of the CONVODE package written for the REDUCE kernel showed some possibility for transforming the problem in two linear first order equations of which the second is a case of the Ricatti equation, at least for $\lambda(r)={\lambda_0r, \lambda_0r^{-1}}$. Is this relevant for other cases?

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Somewhat more generally, for $\lambda(r) = \lambda_0 r^p$, Maple expresses the general solution of the differential equation in terms of Bessel functions of order ${\frac {\sqrt {1+4\,{L}^{2}}}{2\,p+2}}$: $$y \left( r \right) =c_{{1}}\sqrt {r} {{\rm J}\left({\frac {\sqrt {1+4\,{L}^{2}}}{2\,p+2}},\,{\frac {{r}^{p+1}\lambda_{{0}}}{p+1}}\right)} +c_{{2}}\sqrt {r} {{\rm Y}\left({\frac {\sqrt {1+4\,{L}^{2}}}{2\,p+2}},\,{\frac {{r}^{p+1}\lambda_{{0}}}{p+1}}\right)}$$
Meanwhile I have found the particular version for $p=1$ in a slightly different form in Abramowits-Stegun. Simplified versions of the problem are in Wikipedia already. – rtheo Jul 20 '11 at 8:29
Seems that a combination of en.wikipedia.org/wiki/Confluent_hypergeometric_functions would always give an answer apart form some crazy choices of $\lambda$. – rtheo Jul 20 '11 at 8:31