I'm solving the following Schroedinger equation in the domain $r>0$

$\psi''(r) + \left(E-\frac{a}{r^b}\right)\psi(r)=0 $,

where $0 < b < 2$ and $a, E$ are positive constants. Primarily I'm interested in the asymptotical power behavior of the solution as $r\to 0$. To be complete in the description of the problem, I fix my boundary conditions at $r\to +\infty$ as a plane wave ansatz.

I did a lot of DSolve with Mathematica and found out that $\psi(r)\to const$ as $r\to 0$. It gave me a hint for the power series solution, that one of the terms in the expansion

$\psi(r) = \sum\limits_i a_i r^{\alpha_i}$

for some (noninteger) $\alpha_i>0$ should cancel with $a/r^b$. However, even with this assumption there are a bunch of terms which do not cancel. Power expansion does not seem to work in this case as well as the WKB approximation (double checked numerically). What are the other known methods to find an approximate asymptotic behavior in this case? A good reference would be great too!

second orderlinear equation into afirst ordernonlinear one. Maybe this is not simpler, but at least it's different! – Zen Harper Feb 25 '11 at 1:58