## Abstract

We prove asymptotic completeness for operators of the form H = -Δ + L on L^{2}(ℝ^{d}), d ≥ 2, where L is an admissible perturbation. Our class of admissible perturbations contains multiplication operators defined by real-valued potentials V ∈ L^{q}(ℝ ^{d}), q ∈ [d/2, (d + 1)/2] (if d = 2, then we require q ∈ (1, 3/2]), as well as real-valued potentials V satisfying a global Kato condition. The class of admissible perturbations also contains first-order differential operators of the form a · ∇ - ∇ · a for suitable vector potentials a. Our main technical statement is a new limiting absorption principle, which we prove using techniques from harmonic analysis related to the Stein-Tomas restriction theorem.

Original language | English (US) |
---|---|

Pages (from-to) | 397-440 |

Number of pages | 44 |

Journal | Duke Mathematical Journal |

Volume | 131 |

Issue number | 3 |

DOIs | |

State | Published - Feb 15 2006 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)