Lorentzian散射空间上波动算子的复幂与谱作用
Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces
AI总结：
本文在Lorentzian散射空间中用泛函演算定义波动算子复幂，证明迹密度的亚纯性并将极点联系于标量曲率，从而构建Chamseddine-Connes作用玻色部分的Lorentzian谱作用模型。
摘要：
我们考虑Minkowski空间的微扰，以及更一般的、已知波动算子 $\square_g$ 本质自伴的时空。我们通过泛函演算定义复幂 $(\square_g-i\varepsilon)^{-α}$，并证明迹密度作为 $α$ 的亚纯函数存在。我们将其极点与几何量联系起来，特别是标量曲率。这些结果使我们能够表述一种谱作用原理，作为Chamseddine-Connes作用玻色部分的简单Lorentzian模型。我们的证明结合了微局部预解估计（包括径向传播估计）与Hadamard参数矩阵的一致估计。论证直接在Lorentzian号差下进行，不依赖从Euclidean情形转换。结果在超静态时空情形下也成立。
英文摘要：
We consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator $\square_g$ is known to be essentially self-adjoint. We define complex powers $(\square_g-i\varepsilon)^{-α}$ by functional calculus, and show that the trace density exists as a meromorphic function of $α$. We relate its poles to geometric quantities, in particular to the scalar curvature. The results allow us to formulate a spectral action principle which serves as a simple Lorentzian model for the bosonic part of the Chamseddine-Connes action. Our proof combines microlocal resolvent estimates, including radial propagation estimates, with uniform estimates for the Hadamard parametrix. The arguments operate in Lorentzian signature directly and do not rely on a transition from the Euclidean setting. The results hold also true in the case of ultrastatic spacetimes.
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