Question: I stumbled across an exercise in quantum mechanics concerning the self-adjointess of the radial momentum operator. In the exercise, the statement was made that this operator is self-adjoint for wave functions that satisfy

$$\lim\_{r\to0}r\psi\left(r\right)=\lim\_{r\to \infty}r\psi\left(r\right)=0$$

I was wondering, from a pure mathematical point of view, what an example of such a function $\psi\left(r\right)$ would be.

Answer: 
$1/(a^2+r^2)$ will work for $a>0$; the normalisation constant is an exercise.
maybe $\varphi(r)=0$? that's a trivial example.

you can also take $\varphi(r)=2^{1+\varepsilon}$ for $r<1/2$ and $\varphi(r)=1/r^{1+\varepsilon}$ for $r\geq 1/2$ for every $\varepsilon>0$.

In particular, everything that is bounded in an interval around zero and tends to zero faster than $1/r$ as $r$ goes to infinity will work.