If I am considering the space of $H^2(\mathbb{R})$ functions, I have my functions must vanish at infinity by Morrey’s inequality (tell me if I am wrong). Then, we have also if u is real...
\begin{equation} 2\int^{\infty}_{-\infty} u \frac{du}{dx} = [u^2]^{\infty}_{-\infty} \end{equation}
Does the right-hand side equal zero or does it in some framework I am not considering?