An averaging theorem for nonlinear Schr\"odinger equations with small nonlinearities

Consider nonlinear Schr"odinger equations with small nonlinearities $$\frac{d}{dt}u+i(-\mathrm{△}u+V(x)u)=ϵ\mathcal{P}(\mathrm{△}u,u,x),x\in {\mathbb{T}}^d\mathrm{.}\text{eqno}(\ast )$$ Let $\{{\zeta }_1(x),{\zeta }_2(x),\dots \}$ be the $L_2$-basis formed by eigenfunctions of the operator $-\mathrm{△}+V(x)$. For any complex function $u(x)$, write it as \mbox{$u(x)={\sum }_{k⩾1}{v}_k{\zeta }_k(x)$} and set $I_k(u)=\frac{1}{2}\mathrm{\mid }{v}_k{\mathrm{\mid }}^2$. Then for any solution $u(t,x)$ of the linear equation $(\ast {)}_{ϵ=0}$ we have $I(u(t,\cdot ))=const$. In this work it is proved that if $(\ast )$ is well posed on time-intervals $t\lesssim {ϵ}^{-1}$ and satisfies there some mild a-priori assumptions, then for any its solution $u^{ϵ}(t,x)$, the limiting behavior of the curve $I(u^{ϵ}(t,\cdot ))$ on time intervals of order ${ϵ}^{-1}$, as $ϵ\to 0$, can be uniquely characterized by solutions of a certain well-posed effective equation.