## Abstract

In this paper we establish dispersive estimates for solutions to the linear Schrödinger equation in three dimensions (0.1) 1/i∂ _{t}ψ - Δ + Vψ = 0, ψ(s) = f where V(t, x) is a time-dependent potential that satisfies the conditions sup _{t}∥V(t,·)∥_{L3/2(ℝ3)} + sup _{x∈ℝ3}∫ℝ3∫_{-∞} ^{∞}|V(τ̂, x)|/|x - y| dτ dy < c_{0}. Here c_{0} is some small constant and V(τ̂, x) denotes the Fourier transform with respect to the first variable. We show that under these conditions (0.1) admits solutions ψ(·) ∈ L_{t} ^{∞}(L_{x}^{2}(ℝ^{3})) ∩ L _{t}^{2}(L_{x}^{6}(ℝ^{3})) for any f ∈ L^{2}(ℝ^{3}) satisfying the dispersive inequality (0.2) ∥ψ(t)∥_{∞} ≤ C|t - s| ^{-3/2}∥f∥_{1} for all times t,s. For the case of time independent potentials V(x), (0.2) remains true if ∫_{ℝ6} |V(x)| |V(y)|/|x - y|^{2}dxdy < (4π)^{2} and ∥V∥_{K} : = sup _{x∈ℝ3}∫ _{ℝ3} |V(y)|/|x - y|dy < 4π. We also establish the dispersive estimate with an ε-loss for large energies provided ∥V∥_{K} + ∥V∥_{2} < ∞. Finally, we prove Strichartz estimates for the Schrödinger equations with potentials that decay like |x|^{-2-ε} in dimensions n ≥ 3, thus solving an open problem posed by Journé, Soffer, and Sogge.

Original language | English (US) |
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Pages (from-to) | 451-513 |

Number of pages | 63 |

Journal | Inventiones Mathematicae |

Volume | 155 |

Issue number | 3 |

DOIs | |

State | Published - Mar 24 2004 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)