JournalsjstVol. 7, No. 4pp. 1039–1099

Ruled nodal surfaces of Laplace eigenfunctions and injectivity sets for the spherical mean Radon transform in R3\mathbb R^3

  • Mark L. Agranovsky

    Bar-Ilan University, Ramat Gan, Israel
Ruled nodal surfaces of Laplace eigenfunctions and injectivity sets for the spherical mean Radon transform in $\mathbb R^3$ cover
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Abstract

It is proved that if a Paley–Wiener family of eigenfunctions of the Laplace operator in R3\mathbb R^3 vanishes on a real-analytically ruled two-dimensional surface SR3S \subset \mathbb R^3 then SS is a union of cones, each of which is contained in a translate of the zero set of a nonzero harmonic homogeneous polynomial. If SS is an immersed C1C^1 manifold then SS is a Coxeter system of planes. Full description of common nodal sets of Laplace spectra of convexly supported distributions is given. In equivalent terms, the result describes ruled injectivity sets for the spherical mean transform and confirms, for the case of ruled surfaces in R3,\mathbb R^3, a conjecture from [1].

Cite this article

Mark L. Agranovsky, Ruled nodal surfaces of Laplace eigenfunctions and injectivity sets for the spherical mean Radon transform in R3\mathbb R^3. J. Spectr. Theory 7 (2017), no. 4, pp. 1039–1099

DOI 10.4171/JST/185