标题： 波算子的复次幂和洛伦兹散射空间上的谱作用 显示英文标题

标题： Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces

摘要： 我们考虑闵可夫斯基空间的扰动以及更一般的时空，其中波算子$\square_g$已知是本质自伴的。 我们通过函数演算定义复次幂$(\square_g-i\varepsilon)^{-\alpha}$，并证明迹密度作为$\alpha$的亚纯函数存在。 我们将它的极点与几何量相关联，特别是与标量曲率相关。 这些结果使我们能够制定一个谱作用原理，该原理作为Chamseddine-Connes作用的玻色部分的简单洛伦兹模型。 我们的证明结合了微局部预解估计，包括径向传播估计，以及Hadamard参数的均匀估计。 这些论证直接在洛伦兹符号下进行，不依赖于从欧几里得设置的转换。 这些结果在超静态时空的情况下也成立。 显示英文摘要

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 $(\square_g-i\varepsilon)^{-\alpha}$ by functional calculus, and show that the trace density exists as a meromorphic function of $\alpha$. 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.