论文标题
在一维迪拉克方程上具有潜力
On the one dimensional Dirac equation with potential
论文作者
论文摘要
我们将$ l^1 \调查至l^\ infty $的分散估计，具有潜力。特别是，我们表明，狄拉克的演变满足了自然的$ t^{ - \ frac12} $衰减率，当阈值是常规时，可以将其提高到$ t^{ - \ frac32} $，以空间重量为代价。我们对阈值障碍物的结构进行了分类，表明每个阈值最多都有一个维空间。我们表明，在存在阈值共振的情况下，Dirac Evolution满足自然衰减速率，并满足加权速度更快的绑定，除了最多的排名最多，每个阈值一个。此外，我们证明，相对于初始数据所需的平滑性，高能分散界几乎是最佳的。为此，我们使用了最初开发的高能量参数的变体来研究磁性schrödingeroberators的Kato平滑估计。以前从未使用过此方法来获得$ l^1 \至l^\ infty $估计。由于我们的分析，我们证明了统一的限制吸收原理，Strichartz估计并证明了具有非自发接合潜能的一维迪拉克操作员的特征值区域。
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 $t^{-\frac12}$ decay rate, which may be improved to $t^{-\frac32}$ 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.
