Eigenfunction equivalence for the fractional Laplace-Beltrami operator and the classical Helmholtz equation

Abstract

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.