A HOLOGRAPHIC GLOBAL UNIQUENESS IN PASSIVE IMAGING

— We consider a radiation solution ψ for the Helmholtz equation in an exterior region in R³. We show that the restriction of ψ to any ray L in the exterior region is uniquely determined by its imaginary part Im ψ on an interval of this ray. As a corollary, the restriction of ψ to any plane X in the exterior region is uniquely determined by Im ψ on an open domain in this plane. These results have holographic prototypes in the recent work Novikov (2024, Proc. Steklov Inst. Math. 325, 218-223). In particular, these and known results imply a holographic type global uniqueness in passive imaging and for the Gelfand-Krein-Levitan inverse problem (from boundary values of the spectral measure in the whole space) in the monochromatic case. Some other surfaces for measurements instead of the planes X are also considered.