Inverse Steklov Spectral Problem for Curvilinear Polygons

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 [20] 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 factorization theorem, allowing us to reconstruct a trigonometric function from the asymptotics of its roots.

Automated summary

This paper investigates the inverse Steklov spectral problem for curvilinear polygons and shows that under specific conditions, the asymptotics of Steklov eigenvalues can determine the number of vertices, the properly ordered sequence of side lengths, and the angles. Counterexamples are presented for when the generic assumptions fail. Additionally, the paper demonstrates the existence of non-isometric triangles with closely related Steklov spectra and utilizes the Hadamard-Weierstrass factorization theorem and other techniques to reconstruct trigonometric functions from their roots' asymptotics.