REMARKS ON NONLINEAR SCHRODINGER EQUATION WITH MAGNETIC FIELDS

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REMARKS ON NONLINEAR SCHRODINGER EQUATION WITH MAGNETIC FIELDS LAURENT MICHEL Abstract. We study the nonlinear Schrodinger equation with time-depending magnetic field without smallness assumption at infinity. We obtain some re- sults on the Cauchy problem, WKB asymptotics and instability. 1. Introduction We consider the nonlinear Schrodinger equation with magnetic field on Rn, n ≥ 1 (1.1) i∂tu = HA(t) u+ b ?f(x, u) with initial condition (1.2) u|t=t0 = ?. Here HA(t) = n∑ j=1 (i∂xj ? bAj(t, x)) 2, t ? R, x ? Rn is the time-depending Schrodinger operator associated to the magnetic potential A(t, x) = (A1(t, x), . . . , An(t, x)), the parameter b > 0 measures the strength of the magnetic field and ? ≥ 0. We sometimes omit the space dependence and write A(t) instead of A(t, x). The first aim of this note is to study the Cauchy problem in the energy space. At the end of the paper we show how recent improvement in the qualitative study of nonlinear Schrodinger equations can be adapted to the magnetic context. Let us begin with the general framework of our study.

constant magnetic field

schrodinger-maxwell sys- tem

strichartz estimates

depending schrodinger

cauchy problem

fixed constant

without loss

schrodinger equation

magnetic field

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pefav

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¨ REMARKS ON NONLINEAR SCHRODINGER EQUATION WITH MAGNETIC FIELDS LAURENT MICHEL Abstract. WestudythenonlinearSchr¨odingerequationwithtime-depending magnetic ﬁeld without smallness assumption at inﬁnity. We obtain some re-sults on the Cauchy problem, WKB asymptotics and instability.

1. Introduction WeconsiderthenonlinearSchr¨odingerequationwithmagneticﬁeldon R n , n ≥ 1 (1.1) i∂ t u = H A(t) u + b γ f ( x, u ) with initial condition (1.2) u | t = t 0 = ϕ. Here n H A(t) = X ( i∂ x j bA j ( t, x )) 2 , t ∈ R , x ∈ R n − j =1 isthetime-dependingSchro¨dingeroperatorassociatedtothemagneticpotential A ( t, x ) = ( A 1 ( t, x ) , . . . , A n ( t, x )), the parameter b > 0 measures the strength of the magnetic ﬁeld and γ ≥ 0. We sometimes omit the space dependence and write A ( t ) instead of A ( t, x ). The ﬁrst aim of this note is to study the Cauchy problem in the energy space. At the end of the paper we show how recent improvement inthequalitativestudyofnonlinearSchro¨dingerequationscanbeadaptedtothe magnetic context. Let us begin with the general framework of our study. We suppose that the magnetic potential is a smooth function A ∈ C ∞ ( R t × R nx , R n ) and that it satisﬁes the following assumption. Assumption 1. There exists some constants C α > 0 , α ∈ N n such that (1) ∀ α ∈ N n sup | ∂ xα ∂ t A | ≤ C α . ( t,x ) ∈ R × R n (2) ∀| α | ≥ 1 , sup | ∂ xα A | ≤ C α . ( t,x ) ∈ R × R n (3) ∃  > 0 , ∀| α | ≥ 1 , sup | ∂ xα B | ≤ C α h x i − 1 −  ( t,x ) ∈ R × R n where B ( t, x ) is the matrix deﬁned by B jk = ∂ x j A k − ∂ x k A j . Note that compactly supported perturbations of linear (with respect to x ) mag-netic potentials satisfy the above hypothesis. Under Assumption 1, the domain D (H A(t) ) = { u ∈ L 2 ( R nx ) , H A(t) u ∈ L 2 ( R xn ) } does not depend on t . Indeed, for t, t 0 ∈ R one has (1.3) H A(t’) = H A(t) + bW ( t, t 0 )( i r x − bA ( t )) + b ( i r x − bA ( t )) W ( t, t 0 ) + b 2 W ( t, t 0 ) 2 Key words and phrases. NonlinearSchro¨dingerequation,Magneticﬁelds,WKBasymptotics. 1

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