Abstract
The classical inverse problem of the determination of a smooth simply-connected planar domain by its Steklov spectrum [1] is equivalent to the problem of the reconstruction, up to conformal equivalence, a positive function \(a \in C^\infty (\mathbb{S})\) on the unit circle \((\mathbb{S}) = \left\{ {e^{i\theta } } \right\}\) from the spectrum of the operator aΛ e , where Λ e = (−d 2/dθ 2)1/2. We introduce 2k-forms Z k (a) (k = 1, 2,... ) of the Fourier coefficients of a, called the zeta-invariants. These invariants are determined by the eigenvalues of aΛ e . We study some properties of the forms Z k (a); in particular, their invariance under the conformal group. A few open questions about zeta-invariants is posed at the end of the article.
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Original Russian Text Copyright © 2015 Mal’kovich E.G. and Sharafutdinov V.A.
The first author was supported by a Grant of the Government of the Russian Federation for the State Maintenance of Scientific Research (Contract 14.B25.31.0029). The article was started by the second author during his stay at the Mittag-Leffler Institute (Sweden) in January–March 2013 in the framework of the “Inverse Problems” program. The second author expresses his gratitude to the Institute for financial support and hospitality and acknowledge the support of the Russian Foundation for Basic Research (Grant 15–01–05929).
Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 4, pp. 853–877, July–August, 2015; DOI: 10.17377/smzh.2015.56.411.
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Mal’kovich, E.G., Sharafutdinov, V.A. Zeta-invariants of the Steklov spectrum of a planar domain. Sib Math J 56, 678–698 (2015). https://doi.org/10.1134/S0037446615040114
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DOI: https://doi.org/10.1134/S0037446615040114