Communications in Mathematical Physics - Volume 260 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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Introduction In this note we adapt recent results of Burq-G´erard-Tzvetkov [2] and Christ-Colliander-Tao [3] on instability for non-linear Schr¨odinger equations to the semi-classical setting. Rather than work with Sobolev spaces we estimate the sizes of solutions and their differences in terms of the small constant, h, coming from the equation. The ideas remain exactly the same but we gain in the simplicity of the arguments and, we hope, in physical relevance. 1) where the coupling constant g is given in terms of the Planck constant and the scattering length a: 4π 2 a (N − 1) .

For clarity we fix it to be equal to + (defocusing). Let uh be as in the statement of Theorem 1: x − xj j . 4) the solution of j j j ih∂t vh = ah2 |vh |2 vh . 2) for the two ans¨atze. 5) . We also compute dpr (vh1 (t), vh2 (t)) = cos−1 −2 a(φ 2 (x−x / h)−φ 2 (x−x / h)) 1 2 R3 φ(x − x1 / h)φ(x − x2 / h)eith dx . 1. Suppose that φ ∈ Cc∞ (R3 ; R) has a nondegenerate maximum and that φ L2 = 1. Then for σ |y|−1 1, R3 φ(x − y)φ(x)eiσ (φ 2 (x−y)−φ 2 (x)) dx ≤ b + O(|y| + 1/(σ |y|)) , where b < 1 depends only on φ.

A. Nekrasov Commun. Math. Phys. 1007/s00220-005-1402-x Communications in Mathematical Physics Instability for the Semiclassical Non-linear Schr¨odinger Equation Nicolas Burq1,2 , Maciej Zworski3 1 Universit´e Paris Sud, Math´ematiques, Bˆat. 425, 91405 Orsay Cedex, France. fr 2 Institut Universitaive de France 3 Mathematics Department, University of California, Evans Hall, Berkeley, CA 94720, USA. edu Received: 6 August 2004 / Accepted: 17 March 2005 Published online: 2 August 2005 – © Springer-Verlag 2005 Abstract: We adapt recent results on instability for non-linear Schr¨odinger equations to the semi-classical setting.

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