障碍物散射中的迹奇异性及相关迹的Poisson关系
Trace singularities in obstacle scattering and the Poisson relation for the relative trace
摘要：
我们考虑的情况下，散射的几个障碍物在$\mathbb{R}^d$的$d \geq 2$的拉普拉斯算子$Δ$与Dirichlet边界条件施加在障碍物。在两个障碍物的情况下，我们有拉普拉斯算子$Δ_1$和$Δ_2$通过只对其中一个物体施加Dirichlet边界条件而得到。相对迹算子$g(Δ) - g(Δ_1) - g(Δ_2) + g(Δ_0)$在[18]中被引入，并被证明是一大类函数$g$的迹类，包括多项式增长的某些函数。当$g$在零处足够规则并且在无穷远处快速衰减时，则通过Birman-Krein公式，可以从相对谱移函数$ξ_{rel}(λ) = -\frac{1}π \Im(Ξ(λ))$计算该迹，其中$Ξ(λ)$在上半平面中是全纯的并且快速衰减。本文研究了波迹对$ξ_{rel}$傅里叶变换奇异性的贡献。特别是，我们证明了$\hatξ_{rel}$是真正的解析接近零，我们与衰减的$Ξ(λ)$沿虚轴的第一波迹不变量的最短边界球之间的障碍。函数$Ξ(λ)$在物理学中很重要，因为它决定了物体之间的卡西米尔相互作用。
英文摘要：
We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$ for the Laplace operator $Δ$ with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators $Δ_1$ and $Δ_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative trace operator $g(Δ) - g(Δ_1) - g(Δ_2) + g(Δ_0)$ was introduced in [18] and shown to be trace-class for a large class of functions $g$, including certrain functions of polynomial growth. When $g$ is sufficiently regular at zero and fast decaying at infinity then, by the Birman-Krein formula, this trace can be computed from the relative spectral shift function $ξ_{rel}(λ) = -\frac{1}π \Im(Ξ(λ))$, where $Ξ(λ)$ is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of $ξ_{rel}$. In particular we prove that $\hatξ_{rel}$ is real-analytic near zero and we relate the decay of $Ξ(λ)$ along the imaginary axis to the first wave-trace invariant of the shortest bounding ball orbit between the obstacles. The function $Ξ(λ)$ is important in physics as it determines the Casimir interactions between the objects.
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