AI 中文总结
该论文研究混合符号伪黎曼散射流形上拉普拉斯型算子复幂的谱zeta密度，证明其亚纯延拓并将极点与局部几何不变量关联，方法结合费曼预解估计与Hadamard参数化分析。
AI 中文摘要
我们研究混合符号和任意维数的伪黎曼散射空间上主标量拉普拉斯型算子的复幂所关联的谱zeta密度。该构造超越了偶数维洛伦兹情形，扩展了Dang和Wrochna发展的费曼复幂的局部分析。具体而言，对于ε>0，我们考虑复幂(P±iε)^{-α}，并证明对角迹密度允许亚纯延拓。我们将它们的极点与局部几何不变量联系起来，特别是与伪黎曼标量曲率相联系。我们的证明结合了均匀费曼预解估计与伪黎曼符号中Hadamard参数化的直接分析。超越洛伦兹情形，因果凸性的使用被几何无返回论证所取代。
英文摘要
We study spectral zeta densities associated with complex powers of principally scalar Laplace-type operators on pseudo-Riemannian scattering spaces of mixed signature and arbitrary dimension. The construction extends, beyond the even-dimensional Lorentzian case, the local analysis of Feynman complex powers developed by Dang and Wrochna. Namely, for $\varepsilon > 0$, we consider complex powers $(P\pm i\varepsilon)^{-α}$ and prove that the diagonal trace densities admit meromorphic continuation. We relate their poles to local geometric invariants, in particular to the pseudo-Riemannian scalar curvature. Our proof combines uniform Feynman resolvent estimates with a direct analysis of the Hadamard parametrix in pseudo-Riemannian signature. Beyond the Lorentzian case, the use of causal convexity is replaced by a geometric no-return argument.
