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Mathematics > Analysis of PDEs

arXiv:math/0401355 (math)
[Submitted on 26 Jan 2004 (v1), last revised 22 Nov 2004 (this version, v2)]

Title:Scattering Theory for the Schr"odinger Equation in some external time dependent magnetic fields

Authors:J. Ginibre, G. Velo
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Abstract: We study the theory of scattering for a Schr"odinger equation in an external time dependent magnetic field in the Coulomb gauge, in space dimension 3. The magnetic vector potential is assumed to satisfy decay properties in time that are typical of solutions of the free wave equation, and even in some cases to be actually a solution of that equation. That problem appears as an intermediate step in the theory of scattering for the Maxwell-Schr"odinger (MS) system. We prove in particular the existence of wave operators and their asymptotic completeness in spaces of relatively low regularity. We also prove their existence or at least results going in that direction in spaces of higher regularity. The latter results are relevant for the MS system. As a preliminary step, we study the Cauchy problem for the original equation by energy methods, using as far as possible time derivatives instead of spaces derivatives.
Comments: 70 pages. Expanded introduction and minor rewriting of Section 4
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
MSC classes: Primary 35P25; Secondary 35B40; 35Q40; 81U99
Report number: LPT 04-02
Cite as: arXiv:math/0401355 [math.AP]
  (or arXiv:math/0401355v2 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.math/0401355

Submission history

From: Jean Ginibre [view email]
[v1] Mon, 26 Jan 2004 16:51:20 UTC (44 KB)
[v2] Mon, 22 Nov 2004 16:46:43 UTC (46 KB)
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