## Abstract

In this paper we consider Schrödinger operators H = -Δ + i(A · ∇ + ∇ · A) + V = -Δ + L in R^{n}, n ≥ 3. Under almost optimal conditions on A and V both in terms of decay and regularity we prove smoothing and Strichartz estimates, as well as a limiting absorption principle. For large gradient perturbations the latter is not an immediate corollary of the free case as T(λ) := L(-Δ - (λ^{2} + i0))^{-1} is not small in operator norm on weighted L^{2} spaces as λ → ∞. We instead deduce the existence of inverses (I + T(λ))^{-1} by showing that the spectral radius of T (λ) decreases to zero. In particular, there is an integer m such that lim sup_{λ→∞} ∥T(λ) ^{m}∥ < . This is based on an angular decomposition of the free resolvent for which we establish the limiting absorption bound (0.1) ∥D ^{α}ℛ_{d,δ}(λ^{2}) f∥_{B*} ≤ C_{n}λ ^{-1+|α|}∥f∥_{B} where 0 ≤ |α| ≤ 2, B is the Agmon-Hörmander space, and ℛ _{d,δ}(λ^{2}) is the free resolvent operator at energy λ^{2} whose kernel is restricted in angle to a cone of size δ and by d away from the diagonal x = y. The main point is that C_{n} only depends on the dimension, but not on the various cut-offs. The proof of (0.1) avoids the Fourier transform and instead uses Hörmander's variable coefficient Plancherel theorem for oscillatory integrals.

Original language | English (US) |
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Pages (from-to) | 687-722 |

Number of pages | 36 |

Journal | Forum Mathematicum |

Volume | 21 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2009 |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics