Convergence of second derivative in $L^2(\mathbb{R}^+)$

Question

Could you find a sequence $f_n$ of smooth functions with compact support over the half line $\mathbb{R}^+$ such that $f_n$ converges in $L^2(\mathbb{R}^+)$ but such that the second derivatives $f^{\prime\prime}_n$ don't converge in $L^2(\mathbb{R}^+)$?

Answer 1

Sketch: Define

$$\begin {cases} f(x) = e^{1/(x^2-1)},& x\in (-1,1)\\ 0, & |x|\ge 0 \end {cases}$$

Then $f_n(x) = f(nx) \to 0$ in $L^2(\mathbb R),$ but $\|f_n''(x)\|_2 \to \infty.$ Now all of these functions have compact support in $[-1,1]$ so to get an example on $(0,\infty),$ just shift the above two units to the right.