AI 中文总结
本文建立了分数次Laplace-Beltrami算子与经典Helmholtz方程的本征函数等价性，基于Seeley的椭圆算子复幂构造与伪微分符号演算完成证明，并将其应用于固定频率逆散射问题。
AI 中文摘要
本文研究(ℝᵈ,g)上与分数次Laplace-Beltrami方程相关的谱问题，并建立其与经典各向异性Helmholtz方程的等价性。该证明基于Seeley对椭圆算子复幂的构造及伪微分符号演算。作为应用，我们描述了从散射振幅恢复度量的相关固定频率逆散射问题。
英文摘要
In this article we study the spectral problem related to the fractional Laplace-Beltrami equation on $(\mathbb{R}^d,g)$ and establish its equivalence with the classical anisotropic Helmholtz equation. The proof is based on Seeley's construction of complex powers of elliptic operators and the pseudodifferential symbolic calculus. As an application, we describe the related fixed-frequency inverse scattering problem of recovering the metric from the scattering amplitude.
Comments12 pages
