On the one dimensional Dirac equation with potential

Abstract

We investigate $L^1\to L^{\infty }$ dispersive estimates for the one dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural $t^{-\frac{1}{2}}$ decay rate, which may be improved to $t^{-\frac{3}{2}}$ at the cost of spatial weights when the thresholds are regular. We classify the structure of threshold obstructions, showing that there is at most a one dimensional space at each threshold. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natural decay rate, and satisfies the faster weighted bound except for a piece of rank at most two, one per threshold. Further, we prove high energy dispersive bounds that are near optimal with respect to the required smoothness of the initial data. To do so we use a variant of a high energy argument that was originally developed to study Kato smoothing estimates for magnetic Schrödinger operators. This method has never been used before to obtain $L^1\to L^{\infty }$ estimates. As a consequence of our analysis we prove a uniform limiting absorption principle, Strichartz estimates and prove the existence of an eigenvalue free region for the one dimensional Dirac operator with a non-self-adjoint potential.

arXiv commentThis version corrects an error in Lemma~2.2 of the published version which also necessitates changes to the statement of Theorems~1.1 and 2.3 for high energies only. The authors thank Joseph Kraisler, Amir Sagiv and Michael Weinstein for pointing out the error in the published version

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Cited by in corpus (4)

• Keller-type bounds for Dirac operators perturbed by rigid potentials
• Asymptotic stability of small bound state of nonlinear quantum walks
• The Massless Dirac Equation in Three Dimensions: Dispersive estimates and zero energy obstructions
• Decay of solutions of nonlinear Dirac equations