A holographic uniqueness theorem for the two-dimensional Helmholtz equation

2024/08/09 by Nair, Arjun, Novikov, Roman · 1 citation
Computer Science · Engineering · Mathematics · #35J05 #35P25 #35R30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2408.08326

openalex created_date 2024/08/09 · openalex publication_date 2024/08/09 · openalex updated_date 2026/07/28

Abstract

We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region in \mathbb R². We consider a straight line in this region, such that the direction of propagation of the plane wave is not parallel to this line. We show that the radiation solution in the exterior region is uniquely determined by the intensity of the total solution on an interval of this line. In particular, this result solves one of the old mathematical questions of holography in its two-dimensional setting. Our proofs also contribute to the theory of the Karp expansion of radiation solutions in two dimensions.

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