Thank you Jeff, that's very interesting. So one could conjecture that $\sum_{n=1}^\infty n \frac{a_n}{(1+a_n)^2} =\infty$ is a necessary and sufficient condition for the sequence $f_n$ as you defined to span $L^2[0,\infty)$. The that $a_n$ is constant is the most famous one, with the Laguerre functions used to solve the Hydrogen atom. So you gave a very nice proof of the completeness of the Lagueree functions.

@Will: It is not claimed that these function themselves are solutions of a differential equation, but rather that they were used in a variational approximation scheme to approximate eigenvalues of a Schroedinger operator with some potential. I think you are right that the sought-for eigenfunctions were to vanish at the origin, but as far as the question of density in $L^2[0,\infty)$ is concerned, the condition of vanishing at the origin has no meaning since $L^2$ function are only defined almost-everywhere.

Thank you. The connection with complex analysis is very interesting, and I am indeed interested in a general approach to such problems, not only in the specific question.

The above argument would seem to imply, for example, that the sequence $x,x^2,x^3,...$ does not span $L^2[0,1]$ (since all these functions vanish at $0$) - but in fact it does, by simple arguments. The problem is, as pointed out by Philipp, the implication from $L^2$ to pointwise convergence.

Thanks coudy. I tried implementing your suggestion (on MAPLE). The distance from $e^{-x}$ to $F_n$ do seem to decrease very slowly with $n$: for $n=10$ it is $0.209$, for $n=40$ it is $0.201$ (I can't do much higher $n$ because the determinant computation takes a long time). So this at least suggests that $e^{-x}$ is not in the span.

@Andrey: In conversation with a physicist, I was told that someone had used such a sequence for numerically approximating eigenfunctions of a radial Schroedinger operator, and obtained bad results. The explanation offered was that it does not form a basis. I therefore wonder how to show this (if its true).