can anyone give me a reference about the following kind of generalised spherical Bessel equation:
$$\left[x^2\partial_x^2 +2x\partial_x +k^2(x^2+\alpha x)-l(l+1)\right]f(x)=0$$
or for $u(x)=xf(x)$:
$$\left[\partial_x^2+k^2\left(1+\frac{\alpha}{x}\right)-\frac{l(l+1)}{x^2}\right]u(x)=0.$$
I could not find any "modified" or "generalised" form of Bessel's equation that included a linear term like the $k^2\alpha x$ term above.
Thank you.
With $\rho=kx$ and $u(x)=w(\rho)$, this equation becomes the Coulomb wave equation: $$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\rho}^{2}}+\left(1+\frac{ak}{\rho}-% \frac{\ell(\ell+1)}{\rho^{2}}\right)w=0$$ Solutions of the initial equation are then $$ f(x)=\frac{A}{x}F_\ell\left(-\frac{ak}2,kx\right)+\frac{B}{x}F_\ell\left(-\frac{ak}2,kx\right)$$ build from the Coulomb functions.