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I was working on an exercise with a modified Coulumb potential, which could be a good example for anyone trying to expand their experience besides the classical QM examples. Consider the following potential:

$$V(r)=-\frac{Ze^2}{r}\left (1-\frac{\lambda}{r} \right)$$

With $\lambda>0$. This obviously is an attractive Coulumb potential plus an repulsive inverse-square term that effectively produces a minimum on:

$$r_{min}={Ze^2}{\lambda}$$

I tried to obtain the solutions of the radial Schrodinger equation

$$[\frac{-h^2}{2m}\frac{d^2}{dr^2}+\frac{h^2l(l+1)}{2mr^2}+V(r)]u(r)=Eu(r)$$

by analyzing the behaviors on the limits. When $r\rightarrow\infty$, we have:

$$[\frac{-h^2}{2m}]u(r)=Eu(r)$$

With solutions of the form $u(r)=exp{(-\xi r)}$, with $\xi=\sqrt{-2mE/h^2}$.

When $r\rightarrow 0$ we have:

$$\left [\frac{-h^2}{2m}\frac{d^2}{dr^2}+\frac{h^2l(l+1)}{2mr^2}+\frac{Ze^2\lambda}{r^2} \right ]u(r)=0$$

With solutions of the form $u(r)=Ar^k$, with $k=l(l+1)-Ze^2h^2\lambda/2m$. Hence, the solutions have the form:

$$u(r)=Cr^ke^{-\xi r}w(r)$$

Usually when we only have the Coulumb potential we can find the Laguerre polynomials by setting $w(r)$ as a sum of polynomials and cutting the series. However, this is only valid since, in that case, $k=l(l+1)$ is an integer which allows us to express a recurrence relationship.

However, in this case, $k$ isn't necessarily and integer so I'm not sure if I should obtain the same Laguerre polynomials or the solution has another special form. Knowing how to do this could help to generalize for any other modified Coulumb potential.

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Laguerre polynomials are a special case of the more general form of the solution. In general, Coulomb radial equation can be solved in terms of the confluent hypergeometric functions: Kummer's ${}_1F_1$ (also known as $M$) and Tricomi's $U$. The one nonsingular at $r=0$ is ${}_1F_1$.

In the usual hydrogen atom problem, the order parameters of these functions are $-n+l+1,\;2l+2$, which is the case of the usual constraints on $n$ and $l$ result in termination of the hypergeometric series, giving the Laguerre polynomials (see the relation of Laguerre polynomials to hypergeometric functions).

In your problem you simply shouldn't try to get the polynomial form, and instead continue working with the hypergeometric functions.

Relevant derivations of the solutions in terms of the Kummer's hypergeometric function can be found in e.g. Landau&Lifshitz Quantum Mechanics, $\S36$ "Motion in a Coulomb field (spherical polar coordinates)".

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