# A special sequence

I m looking for a sequence $$(f_j)\in C^\infty(\Bbb{R})$$ such that
$$\int^\infty_0\Big|\partial^2_r f_j+\frac{1}{r}\partial_r f_j+r^2f_j\Big|^2rdr\to 0,$$ and

$$\int_{\Bbb{R^+}}|f_j(r)|^2 rdr=1\quad\forall j\in\Bbb{N}.$$

Could anybody help? Thanks in advance.

• Any squence of $C^\infty$ function with supports on $]-\infty, 0]$ and with weighted $L^2$ norm 1 verifies this. Maybe this site is better suited for your needs : math.stackexchange.com – Bleuderk Feb 27 at 18:11
• @Bleuderk. Thanks for your help, in fact my hypotesis is $\int_{\Bbb{R^+}}|f_j(r)|^2 rdr=1$ not $\int_{\Bbb{R}}|f_j(r)|^2 rdr=1$. – Kacdima Feb 27 at 20:04
• Have you checked this thread on MSE ? – Bleuderk Feb 27 at 22:15
• Yeah, but it does not work. – Kacdima Feb 28 at 10:40
• Could you explain why ? – Bleuderk Feb 28 at 11:06