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.

  • $\begingroup$ 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 $\endgroup$ – Bleuderk Feb 27 at 18:11
  • $\begingroup$ @Bleuderk. Thanks for your help, in fact my hypotesis is $\int_{\Bbb{R^+}}|f_j(r)|^2 rdr=1$ not $\int_{\Bbb{R}}|f_j(r)|^2 rdr=1$. $\endgroup$ – Kacdima Feb 27 at 20:04
  • $\begingroup$ Have you checked this thread on MSE ? $\endgroup$ – Bleuderk Feb 27 at 22:15
  • $\begingroup$ Yeah, but it does not work. $\endgroup$ – Kacdima Feb 28 at 10:40
  • $\begingroup$ Could you explain why ? $\endgroup$ – Bleuderk Feb 28 at 11:06

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