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Proceedings of the American Mathematical Society

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Transport in the one-dimensional Schrödinger equation

Author: Michael Goldberg
Journal: Proc. Amer. Math. Soc. 135 (2007), 3171-3179
MSC (2000): Primary 35Q40; Secondary 34L25
Published electronically: May 10, 2007
MathSciNet review: 2322747
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Abstract: We prove a dispersive estimate for the Schrödinger equation on the real line, mapping between weighted $ L^p$ spaces with stronger time-decay ( $ \vert t\vert^{-\frac32}$ versus $ \vert t\vert^{-\frac12}$) than is possible on unweighted spaces. To satisfy this bound, the long-term behavior of solutions must include transport away from the origin. Our primary requirements are that $ \langle x\rangle^{3}V$ be integrable and $ -\Delta + V$ not have a resonance at zero energy. If a resonance is present (for example, in the free case), similar estimates are valid after projecting away from a rank-one subspace corresponding to the resonance.

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Michael Goldberg
Affiliation: Department of Mathematics, Johns Hopkins University, 3400 N. Charles St., Baltimore, Maryland 21218

Keywords: Schr\"odinger equation, dispersive estimates, transport, Jost solutions, scattering theory
Received by editor(s): June 12, 2006
Published electronically: May 10, 2007
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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