Limiting absorption principle and Strichartz estimates for Dirac operators in two and higher dimensions

Abstract

In this paper we consider Dirac operators in ${\mathbb{R}}^n$, $n\ge 2$, with a potential $V$. Under mild decay and continuity assumptions on $V$ and some spectral assumptions on the operator, we prove a limiting absorption principle for the resolvent, which implies a family of Strichartz estimates for the linear Dirac equation. For large potentials the dynamical estimates are not an immediate corollary of the free case since the resolvent of the free Dirac operator does not decay in operator norm on weighted $L^2$ spaces as the frequency goes to infinity.

Updated Corollary 1.3 with a slightly stronger statement. To appear in Comm. Math. Phys. arXiv admin note: text overlap with arXiv:0705.0546

References in corpus (3)

• The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions
• Nonrelativistic asymptotics of solitary waves in the Dirac equation with the Soler-type nonlinearity
• On the Global Limiting Absorption Principle for Massless Dirac Operators

Cited by in corpus (7)

• Absence of eigenvalues of Dirac and Pauli Hamiltonians via the method of multipliers
• Eigenvalue bounds for non-selfadjoint Dirac operators
• Kato smoothing, Strichartz and uniform Sobolev estimates for fractional operators with sharp Hardy potentials
• On the one dimensional Dirac equation with potential
• Keller-type bounds for Dirac operators perturbed by rigid potentials
• Time-frequency analysis of the Dirac equation
• The Massless Dirac Equation in Three Dimensions: Dispersive estimates and zero energy obstructions