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.

我们调查 ${升}^1\to {升}^{\infty }$具有势能的一维狄拉克方程的色散估计。特别是，我们证明狄拉克进化满足自然${吨}^{-\frac{1}{2}}$ 衰减率，可以改进为 ${吨}^{-\frac{3}{2}}$当阈值规则时，以空间权重为代价。我们对门槛障碍物的结构进行分类，表明每个门槛最多有一个一维空间。我们表明，在存在阈值共振的情况下，狄拉克演化满足自然衰减率，并满足更快的加权界限，除了最多两个秩，每个阈值一个。此外，我们证明了高能量色散边界对于初始数据所需的平滑度接近最佳。为此，我们使用高能论证的变体，该变体最初是为了研究磁薛定谔算子的 Kato 平滑估计而开发的。以前从未使用过这种方法来获得${升}^1\to {升}^{\infty }$估计。作为我们分析的结果，我们证明了统一的极限吸收原理，Strichartz 估计，并证明了具有非自伴随势的一维狄拉克算子的特征值自由区域的存在。