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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.

m.a.137
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1 Answers1

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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.

Paul Enta
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