Picture below is from the 56th page of Lyapunov stability of ground states of nonlinear dispersive evolution equations. The nonlinear Schrodinger equation (NLS) is $$ i\phi_t(x,t)+\Delta \phi(x,t)+ f(|\phi(x,t)|^2)\phi(x,t)=0 $$ and $R$ is solution (ground state? ) of $$ \Delta R -ER+f(|R|^2)R=0 $$ I fail to calculate the $L_+$ and $L_-$. What I do:
Denote $$ K(\phi)=i\phi_t(x,t)+\Delta \phi(x,t)+ f(|\phi(x,t)|^2)\phi(x,t) $$ then the linearization about the ground state is $$ \frac{d}{d\delta}|_{\delta=0}K(R+\delta \psi)=i\psi_t +\Delta \psi + 2f(|R|^2)\psi+f'(|R|^2)R^2\psi $$ but I don't understand where is imaginary parts, where is real...
How should I do ?
