Localized exterior Calderón uniqueness for complete Bernstein functions of the Laplacian

Abstract

We establish uniqueness in an exterior Calderón problem for nonlocal Schrödinger operators Φ(−Δ)+q on bounded Lipschitz domains of Rn, n⩾2. The unknown is a real-valued potential q∈L∞, and the data are localized exterior Dirichlet-to-Neumann measurements. We identify an explicit class ACBF of complete Bernstein functions such that, for arbitrary nonempty open sets W1,W2 in the unbounded exterior component, equality of the measurements generated and tested in W1 and W2 implies equality of the potentials. The sets may be disjoint. The admissible symbols need not be homogeneous or of a fixed fractional order; the class includes fractional powers, finite positive sums and compact mixtures of powers, logarithmic symbols and their compact mixtures, and relativistic stable symbols. The main new ingredient is an exterior unique continuation principle obtained from a separated-support jump-kernel identity, off-diagonal analyticity of subordinated pure-jump kernels, the Paley–Wiener theorem, and a branch-rigidity argument excluding global meromorphic continuation of Φ. Runge approximation and the Alessandrini identity then yield exact identifiability of q. In the subordinate-diffusion interpretation, Φ specifies a multiscale jump law and q acts as an absorption or reaction coefficient, so the result provides a uniqueness mechanism for remote excitation and observation outside an inaccessible region.