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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.
We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions of curvilinear polygons in terms of their side lengths and angles. These formulae are quite precise: the errors tend to zero as the spectral parameter tends to infinity. The
Let us say that an $n$-sided polygon is semi-regular if it is circumscriptible and its angles are all equal but possibly one, which is then larger than the rest. Regular polygons, in particular, are semi-regular. We prove that semi-regular polygons a
We solve the inverse spectral problem associated with periodic conservative multi-peakon solutions of the Camassa-Holm equation. The corresponding isospectral sets can be identified with finite dimensional tori.
For the Schrodinger equation $-d^2 u/dx^2 + q(x)u = lambda u$ on a finite $x$-interval, there is defined an asymmetry function $a(lambda;q)$, which is entire of order $1/2$ and type $1$ in $lambda$. Our main result identifies the classes of square-in
In this paper, we study the bounds for discrete Steklov eigenvalues on trees via geometric quantities. For a finite tree, we prove sharp upper bounds for the first nonzero Steklov eigenvalue by the reciprocal of the size of the boundary and the diame