Deterministic and stochastic inverse problems for the fractional Schrödinger equation

We are interested in an inverse problem for the fractional Schrödinger equation with a time-independent potential on closed Riemannian manifolds. The model is given by i ∂ t α u + ( ( − Δ g ) β + V ) u = f , align*i∂ₜ^α u+((-Δg)^β+V)u=f, align* with α , β ∈ ( 0 , 1 ) α ,β ∈ (0,1) alpha comma beta element of left parenthesis 0 comma 1 right parenthesis and V ∈ L ∞ ( M ) V∈ L^∞ (M) upper V element of upper L Superscript infinity Baseline left parenthesis upper M right parenthesis real-valued. Our main goal is to determine the fractional powers α , β α ,β alpha comma beta , the Riemannian metric g (up to an isometry), and the potential V simultaneously from the knowledge of the associated source-to-solution map. For disjoint source and observation regions, the map determines α α alpha , β β beta , and the complete cross-spectral data of