论文标题
具有分析性迅速振荡电势的非自我辅助狄拉克运算符的半经典WKB问题
Semiclassical WKB Problem for the non-self-adjoint Dirac operator with an analytic rapidly oscillating potential
论文作者
论文摘要
在本文中，我们研究了非自我辅助狄拉克运算符的散射数据的半经典行为，其迅速振荡电位在实际线的某些社区中是复杂的分析。我们的一些结果是严格且相当一般的。另一方面，完整而具体的理解需要研究特定示例的WKB几何形状。对于此类详细的计算，我们使用了一个特定的示例，该示例已在20年前由Bronski和Miller在数字上进行了研究，并严重依赖其数值计算。我们主要采用确切的WKB方法，为Bohr-Sommerfeld条件提供了完全严格的均匀的半经典分析，用于跨分析弧和相关规范常数的特征值的位置。对于频谱平面中0附近的反射系数以及特征值，我们采用了Olver详细开发的较旧理论。我们的分析是出于需要理解与初始数据$ a \ exp \ {is/ε\} $的聚焦立方NLS方程的半经典行为的动机，这是鉴于Zakharov和Shabat发现的众所周知的事实，即dirac操作员的频谱分析可以通过反向分散的散布理论的NLS方程解决方案。
In this paper we examine the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a rapidly oscillating potential that is complex analytic in some neighborhood of the real line. Some of our results are rigorous and quite general. On the other hand, complete and concrete understanding requires the investigation of the WKB geometry of specific examples. For such detailed computations we use a particular example that has been investigated numerically more than 20 years ago by Bronski and Miller and rely heavily on their numerical computations. Mostly employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues across unions of analytic arcs as well as the associated norming constants. For the reflection coefficient as well as the eigenvalues near 0 in the spectral plane, we employ instead an older theory that has been developed in great detail by Olver. Our analysis is motivated by the need to understand the semiclassical behaviour of the focusing cubic NLS equation with initial data $A\exp\{iS/ε\}$, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator enables the solution of the NLS equation via inverse scattering theory.
