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arXiv · 2004.03881

Inverse Steklov spectral problem for curvilinear polygons

Stanislav Krymski·Michael Levitin·Leonid Parnovski·Iosif Polterovich·David A. Sher
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Abstract

This paper studies the inverse Steklov spectral problem for curvilinear polygons. For generic curvilinear polygons with angles less than $\pi$, we prove that the asymptotics of Steklov eigenvalues obtained in arXiv:1908.06455 determines, in a constructive manner, the number of vertices and the properly ordered sequence of side lengths, as well as the angles up to a certain equivalence relation. We also present counterexamples to this statement if the generic assumptions fail. In particular, we show that there exist non-isometric triangles with asymptotically close Steklov spectra. Among other techniques, we use a version of the Hadamard--Weierstrass factorisation theorem, allowing us to reconstruct a trigonometric function from the asymptotics of its roots.

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BibTeXRIS

Stanislav Krymski, Michael Levitin, Leonid Parnovski, Iosif Polterovich, David A. Sher. 2020-04-08. Inverse Steklov spectral problem for curvilinear polygons. https://doi.org/10.1093/imrn%2Frnaa200

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