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Uniqueness results in the inverse spectral Steklov problem. (English) Zbl 1441.58008

This paper contains some uniqueness results in the inverse spectral Steklov problem. More precisely, it is devoted to an inverse Steklov problem for a particular class of \(n\)-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. The author proves that the knowledge of the Steklov spectrum determines uniquely the associated warping function up to a natural invariance.
The paper is organized as follows. The first section is an introduction to the subject. The author introduces the Calderon and Steklov problems and gives the main results of the paper. The second section is divided in three parts. The first one contains some statements about the separation of variables. The second part deals with the Weyl-Titchmarsh functions. The third part tackles the link between the Dirichlet-to-Neumann (DN) map and the Weyl-Titchmarsh functions. The third section deals with a characterisation by the trace and the determinant. Finally, the forth section contains some uniqueness results on the trace and the determinant.

MSC:

58C40 Spectral theory; eigenvalue problems on manifolds
58J53 Isospectrality
35P20 Asymptotic distributions of eigenvalues in context of PDEs
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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