Helmholtz方程非光滑焦散线的几何和微扰灵敏度
Geometry and Perturbative Sensitivity of non-Smooth Caustics of the Helmholtz Equation
摘要：
利用Fock-Schwinger本征时间公式分析了Helmholtz方程解中非光滑$A_{n>2}$焦散线的几何性质。在本说明书中，$A_3$或尖焦散与复固有时间平面中称为爱因斯坦作用量的极点密切相关。极点的残基在空间曲线上消失，称为鬼源，尖点与鬼源相连。尖点沿鬼源的位置与极点的大小有关。一个类似的地图，提出了有关的einbein行动的基本奇点高阶焦散。的奇异性是源于退化的某一狄利克雷问题的einbein是不同的。因此，爱因斯坦作用量的奇点，以及焦散几何的相关方面，对于折射率的大类扰动是不变的。
英文摘要：
The geometry of non-smooth $A_{n>2}$ caustics in solutions of the Helmholtz equation is analyzed using a Fock-Schwinger proper-time formulation. In this description, $A_3$ or cusp caustics are intimately related to poles of a quantity called the einbein action in the complex proper-time, or einbein, plane. The residues of the poles vanish on spatial curves known as ghost sources, to which cusps are bound. The positions of cusps along the ghost sources is related to the value of the poles. A similar map is proposed to relate essential singularities of the einbein action to higher order caustics. The singularities are shown to originate from degenerations of a certain Dirichlet problem as the einbein is varied. It follows that the singularities of the einbein action, along with the associated aspects of caustic geometry, are invariant with respect to large classes of perturbations of the index of refraction.
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