Picture below is from Oh, Yong-Geun, Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials, Commun. Math. Phys. 121, No. 1, 11-33 (1989). ZBL0693.35132.
I don't know how to get $\langle u_h, u\rangle =0$. What I do :
In fact , $\Psi(x,t)= u_h + u +iv$, I want to use $\frac{d}{dt}\int |\Psi(x,t)|dx=0$ to show (24), but seemly, it is useless.
