An inverse problem for the space-time fractional Schr\"odinger equation on closed manifolds

TL;DR

This paper addresses an inverse problem for a space-time fractional Schrödinger equation on closed manifolds, aiming to recover fractional powers and the Riemannian metric from source-to-solution data.

Contribution

It introduces a novel approach combining asymptotic analysis, eigenvalue Weyl's law, and unique continuation to determine both fractional parameters and geometry from minimal data.

Findings

Successfully determines fractional powers and metric up to isometry

Utilizes Mittag-Leffler functions and eigenvalue asymptotics for analysis

Provides a probabilistic formulation of the inverse problem

Abstract

We formulate an inverse problem for an uncoupled space-time fractional Schr\"odinger equation on closed manifolds. Our main goal is to determine the fractional powers and the Riemannian metric (up to an isometry) simultaneously from the knowledge of the associated source-to-solution map. Our argument relies on the asymptotic behavior of Mittag-Leffler functions, Weyl's law for the eigenvalues of the Laplace-Beltrami operator, the unique continuation property of the space-fractional operator and the disjoint data metric determination result for the wave equation. We also provide a probabilistic formulation of our inverse problem.