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On the wave operators for the critical nonlinear Schr\"odinger equation

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  • 35-Xx Partial Differential Equations


ON THE WAVE OPERATORS FOR THE CRITICAL NONLINEAR SCHRO¨DINGER EQUATION RE´MI CARLES AND TOHRU OZAWA Abstract. We prove that for the L2-critical nonlinear Schro¨dinger equations, the wave operators and their inverse are related explicitly in terms of the Fourier transform. We discuss some consequences of this property. In the one- dimensional case, we show a precise similarity between the L2-critical nonlinear Schro¨dinger equation and a nonlinear Schro¨dinger equation of derivative type. 1. Introduction We consider the defocusing, L2-critical, nonlinear Schro¨dinger equation (1.1) i∂tu+ 1 2 ∆u = |u|4/nu, (t, x) ∈ R× Rn. We consider two types of initial data: Asymptotic state: U0(−t)u(t) ∣∣ t=±∞ = u±, where U0(t) = e i t2∆.(1.2) Cauchy data at t = 0 : u|t=0 = u0.(1.3) It is well known that for data u±, u0 ∈ Σ = H1 ∩ F(H1), where Ff(ξ) = f̂(ξ) = 1 (2pi)n/2 ∫ Rn f(x)e−ix·ξdx, (1.1)–(1.2) has a unique, global, solution u ∈ C(R; Σ) ([GV79], see also [Caz03]). Its initial value u|t=0 is the image of the asymptotic state under the action of the wave operator: u|t=0 =W±u±. Similarly, (1.1)–(1.3) possesses asymptotic states: ∃u± ∈ Σ, ‖U0(−t)u(t)− u±‖Σ −→ t→±∞ 0 : u± =W −1 ± u0. Global well-posedness properties show that the wave operators are homeomor- phisms on Σ. Besides this point, very few properties of these operators are known. The main result of this paper (proved in §2) shows that the wave operators and their inverses are easily related in terms of the Fourier transform: Theorem 1.1. Let n > 1. The following identity holds on Σ: (1.4) F ◦W−1± =W∓ ◦ F . In particular, if C denotes the conjugation f 7→ f , then we have: (1.5) W−1± = (CF)−1W± (CF) . 2000 Mathematics Subject Classification. 35B33; 35B40; 35Q55. 1 2 R. CARLES AND T. OZAWA Using continuity properties of the flow map associated to (1.1), we infer the following result in §3: Corollary 1.2. The result of Theorem 1.1 still holds when Σ is replaced • Either by F(H1), • Or by a neighbo

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