Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Transport in the one-dimensional Schrödinger equation

Author(s): Michael Goldberg
Journal: Proc. Amer. Math. Soc. 135 (2007), 3171-3179.
MSC (2000): Primary 35Q40; Secondary 34L25
Posted: 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 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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