The fractional Calder\'on problem on finite Cayley graphs

Abstract

We study a Calder\'on-type inverse problem for the spectral fractional Laplacian on finite Cayley graphs. For a known self-adjoint operator, recovery from a single measurement reduces to two requirements: full rank of the boundary (exterior)-to-interior coupling and nonvanishing of the interior solution. We verify both requirements on Cartesian products of arbitrary finite groups with even cycles, using consecutive cyclic halves as the accessible and inaccessible sets. Fractionalization increases the coupling rank, and one fixed nonnegative nonzero input determines every nonnegative potential. Our argument relies on a discrete maximum principle and an explicit resolvent formula.