An averaging theorem for nonlinear schrödinger equations with small nonlinearities

Consider nonlinear Schrd̈inger equations with small nonlinearities d d/tu + i(-δu + V (x)u) =εP(δu; δu; u; x); x σ T^d: (*)Let {l1(x);l2(x). g be the L₂-basis formed by eigenfunctions of the operator -δ + V (x). For any complex function u(x), write it as u(x) =σκ≥1 vklk(x)and set Ik(u) = 1/2/vk/2. Then for any solution u(t, x) of the linear equation (*) ε=0 we have I(u(t; ·)) = const. In this work it is proved that if (*) is well posed on time-intervals t ≲ε1 and satisfies there some mild a-priori assumptions, then for any its solution uε (t, x), the limiting behavior of the curve I(uε(t,·)) on time intervals of order ε ^-1, as ε → 0, can be uniquely characterized by solutions of a certain well-posed effective equation.