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Asymptotic evolution for the semiclassical Nonlinear Schr¨odinger equation in presence of electric and magnetic fields Alessandro Selvitella 3 ottobre 2007 ABSTRACT In this paper we study the semiclassical limit for the solutions of a subcritical focusing NLS with electric and magnetic potentials. We consider in particular the Cauchy problem for initial data close to solitons and show that, when the Planck constant goes to zero, the motion shadows that of a classical particle. Several works were devoted to the case of standing waves: differently from these we show that, in the dynamic version, the Lorentz force appears crucially. 1 Introduction We consider the following Cauchy problem for the Nonlinear Schr¨odinger equation (NLS): (t, x) ∈ R × Rk , iut = 21 (ε ∇i − A(x))2 u + V (x)u − |u|p−1 u (CP ) u(x, 0) = φ(x) x ∈ Rk , where (ε ∇ ∇ ε − A(x))2 := −ε2 ∆ + |A(x)|2 − 2εA(x) − ∇ · A(x) i i i (1) is the gauge invariant Laplacian. Here p > 1, k ≥ 2, i is the imaginary unit, V : Rk → R is a smooth scalar function standing for the electric potential, and A : Rk → Rk is a smooth vector function representing a magnetic potential. (CP ) models the motion of a quantum particle moving under the effect of the electric field E := −∇V and the magnetic field B := (Bl,j )l,j := Alxj − Ajxl with j, l = 1 . . . k. We are interested in particular in the semiclassical limit, namely when the Planck constant ε = ~ tends to zero: we will show that, indeed, for initial data of the type: x − X0 i V0 ·x )e , (2) ε where s is a standing wave (see below and (17) ), we recover the motion of a classical particle, precisely: φε (x) ' s( ¨ = −DV (X) + X˙ ∧ B, X P0 ˙ X(0) = X0 , X(0) = = V0 . M (3) NLS in the semiclassical limit is a topic which has received a lot of attention: one of the pioneering works in this field is by Fl