I'm new to this site, but I felt the need to post when I recently came in to an ordinary differential equation/boundary value problem with this form:

$(1)- \frac{d^2 y}{d x^2} + \frac{m(m+1)} {x^2(1 + i \alpha e^{\beta x})} y = k^2 y$

with the boundary conditions $y'(a) = 0$ and $ y'(b) = 0 $. All the constant terms ${\alpha, \beta, m}$ are positive real numbers. The differential operator induced, I'll call $L$, where

$(2) L= - \frac{d^2 }{d x^2} + \frac{m(m+1)} {x^2(1 + i \alpha e^{\beta x})} $

acts on the Hilbert space $L_2(a,b)$. And $0<a<x<b$, that is, $x$ defined on a positive real interval.

The problem I'm experiencing stems from the fact that (1) is a *non*-self adjoint SL problem; where as self-adjoint problems are usually easier and the theory is well known. I'm trying to prove existence and uniqueness of a solution to (1) and know as much as I can about its spectrum and (hopefully) find a rigorous or even a numerical solution (know the eigenvalues/eigenfunctions). But, to be honest, I'm new to functional analysis and advanced differential equations need some advice.

So far, I've been able to write a *formal* solution to (1) using the Frobenius method, by rewriting (1) as

$(3)-{x^2(1 + i \alpha e^{\beta x})} \frac{d^2 y}{d x^2} + \left( m(m+1) - {x^2(1 + i \alpha e^{\beta x})} k^2 \right) y = 0$

and expanding the coefficients in (3) in a Taylor series at $x=0$. This is not very useful though, the recursion formula is very messy and it's very difficult to tell where this solution converges. Another option is to note that the coefficient of $y$ in (1) is infinitely differentiable and the derivatives can be written recursively. That is, I can expand (1) in a Taylor series as well and use Frobenius, and get a different series to work with, which will likely not be much easier. But that's again going to result in a messy recursion relation and (I presume) not be useful.

At this point, I'm a little lost and would really appreciate some professional advice. Again, I'm looking for an existence/uniqueness proof, some theorems or analysis regarding the spectrum and a method for solution (numerical is okay) for the eigenvalues/functions other than what I've mentioned (admittedly, a tall order). I should probably point out that this equation was the result of a separtion of variables of a partial differential boundary value problem in spherical coordinates. The original PDE looked similar to the Helmholtz equation. That's why the ODE above looks similar to the spherical version of Bessel's equation (e.g., set $\alpha=0$).

Regards.