Transport in the one-dimensional Schrödinger equation

Goldberg, Michael

doi:10.1090/S0002-9939-07-08897-1

Transport in the one-dimensional Schrödinger equation
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by Michael Goldberg PDF

Proc. Amer. Math. Soc. 135 (2007), 3171-3179 Request permission

Abstract:

We prove a dispersive estimate for the Schrödinger equation on the real line, mapping between weighted $L^p$ spaces with stronger time-decay ($|t|^{-\frac 32}$ versus $|t|^{-\frac 12}$) 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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