Construction of a sequence Can I find a sequence $(f_j)_{j\in\Bbb{N}}\in  C^{\infty}(\Bbb{R^+})$ such that : $$
 \lim_{j\to\infty}\int^\infty_0 \big(\partial^2_r  f_j+\frac{1}{r}\partial_r f_j+r^2f_j\big)^2 rdr=0$$  and $$ \sqrt{\int_{\Bbb{R^+}}|f_j(r)|^2 rdr}=1 $$ .
Thank you for any suggestion whatsoever.
 A: There are many sequences verifying the conditions. Assuming, for example,
\begin{equation}
f_j=a_je^{-k_jr^2}
\end{equation} 
with $\Re k_j>0$, one has
\begin{equation}
F_j(r)=f''_j(r)+\frac{1}{r}f'_j(r)+r^2f_j(r)=a_j\left(r^2\left( 1+4k_j^2 \right)-4k_j \right)
\end{equation} 
and thus
\begin{equation}
\int_0^\infty\left[F_j(r)\right]^2r\,dr=\frac{a_j^2}{8k_j^3}\left( 16k_j^4+1 \right)
\end{equation} 
One can choose the condition
\begin{equation}
k_j^4\to -\frac{1}{16} \tag{1}\label{1}
\end{equation} 
for $j\to\infty$. 
Also, if $k_j=u_j+iv_j$, with $u_j>0$ and $v_j\in \mathbb{R}$,
\begin{equation}
\left|f_j(r)\right|^2=\left|a_j\right|^2e^{-2u_jr^2}
\end{equation} 
We deduce
\begin{equation}
\int_0^\infty\left|f_j(r)\right|^2r\,dr=\frac{\left|a_j\right|^2}{4u_j}
\end{equation} 
The condition reads 
$$\left|a_j\right|^2=4u_j \tag{2}\label{2}$$ 
Both conditions \eqref{1} and \eqref{2} are satisfied if we choose, for example,
\begin{align}
k_j&=\frac{1+i(1+1/j)}{2\sqrt{2}}\\
a_j&=2^{\frac{1}{4}}
\end{align}
obtaining thus 
\begin{equation}
f_j(r)=2^{\tfrac 1 4}\exp\left(-\tfrac{1+i\left( 1+\frac{1}{j} \right)}{2\sqrt{2}}r^2\right)
\end{equation} 
