Deterministic and stochastic inverse problems for the fractional Schrödinger equation

Abstract 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 , $$\begin{align*}i\partial_t^\alpha u+\big((-\Delta_g)^\beta+V\big)u=f, \end{align*}$$ with α , β ∈ ( 0 , 1 ) $\alpha ,\beta \in (0,1)$ alpha comma beta element of left parenthesis 0 comma 1 right parenthesis and V ∈ L ∞ ( M ) $V\in L^\infty (M)$ 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 $\alpha ,\beta $ 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 $\alpha $ alpha , β $\beta $ beta , and the complete cross-spectral data of