## Abstract

We investigate L ^{1}(ℝ^{4}) → L ^{∞}(ℝ^{4}) dispersive estimates for the Schrödinger operator H = - Δ +V when there are obstructions, a resonance or an eigenvalue, at zero energy. In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator F _{t} satisfying {pipe}F _{t}{pipe}_{L1→L∞} ≲ 1/log t for t > 2 such that (Formula presented.) We also show that the operator F _{t} = 0 if there is an eigenvalue but no resonance at zero energy. We then develop analogous dispersive estimates for the solution operator to the four dimensional wave equation with potential.

Original language | English (US) |
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Pages (from-to) | 1936-1964 |

Number of pages | 29 |

Journal | Communications in Partial Differential Equations |

Volume | 39 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2014 |

## Keywords

- Dispersive estimate
- Resonance
- Schrödinger operator
- Wave equation

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics