Consider the operator $T=-\Delta_{\Bbb{R}^2}-(x^2+y^2)$.
How can I construct a sequence $ (u_n)\in D(T)=\{u\in L^2(R^2); Tu\in L^2(R^2)\}$ such that $\|u_n\|=\sqrt{\int_{R^2}|u(x,y)|^2}=1,$ $u_n\to 0$ weakly, and $\lim_n \|Tu_n\|=0.$
Note that such sequence exists because 0 belongs in the spectrum of $T$ noted $\sigma(T)$ and by the weyl's criterion.