2024/05/12 by Novikov, Roman
#35J05 #35J08 #35P25 #35R30 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2405.10333
We consider a radiation solution ψ for the Helmholtz equation in an exterior region in \mathbb R3. We show that the restriction of ψ to any ray L in the exterior region is uniquely determined by its imaginary part \Imψ 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 \Imψ 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.