The Calderón problem for the multi-fractional Laplacian Schrödinger equation with drift terms
This paper studies the Calderón inverse problem for the multi-fractional Laplacian Schrödinger equation with drift terms, whose leading operator is a sum of fractional Laplacians of different orders, perturbed by a drift vector field and a potential. We consider two inverse settings: in the homogeneous source case \(f=0\), we aim to simultaneously recover the drift coefficient \(b\) and the potential \(q\) from the associated Dirichlet-to-Neumann map; in the non-homogeneous case \(f\neq 0\), we aim to simultaneously determine \(b\), \(q\), and the source term \(f\). Under suitable assumptions, we establish the well-posedness of the forward problem, derive a strong unique continuation principle, prove Runge-type approximation properties in fractional Sobolev spaces and weighted \(C^k\) spaces, and obtain generalized Alessandrini identities. These tools yield global uniqueness results for the inverse problems. In addition, we develop a numerical reconstruction framework combining finite difference discretization, radial basis function surrogate modeling, and hierarchical Bayesian inversion. Numerical examples in one dimension for smooth and discontinuous coefficients and sources illustrate the effectiveness of the proposed method and support the theoretical uniqueness results.