Consider the following operator $H=\partial_{x^2}+x^2$ acting on $L^2(\Bbb{R})$. It's well known that the spectrum of $H$ is $\Bbb{R}$. So by weyl's criteron there exists $(u_n)\in D(H)$ such that
$||u_n||=1$ and $H(u_n)\to 0$ in $L^2(\Bbb{R})$.
Can I find explicitly such sequence?