## Abstract

The inverse problem of recovering a smooth simply connected multisheet planar domain from its Steklov spectrum is equivalent to the problem of determination, up to a gauge transform, of a smooth positive function a on the unit circle from the spectrum of the operator aΛ, where Λ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by Z_{m}(a)=Tr[(aΛ)^{2m}−(aD)^{2m}] for every smooth function a. In the case of a positive a, zeta-invariants are determined by the Steklov spectrum. We obtain some estimate from below for Z_{m}(a) in the case of a real function a. On using the estimate, we prove the compactness of a Steklov isospectral family of planar domains in the C^{∞}-topology. We also describe all real functions a satisfying Z_{m}(a)=0.

Original language | English |
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Pages (from-to) | 1712-1755 |

Number of pages | 44 |

Journal | Journal of Functional Analysis |

Volume | 275 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1 Oct 2018 |

## Keywords

- Dirichlet-to-Neumann operator
- Inverse spectral problem
- Steklov spectrum
- Zeta function
- SETS
- NEUMANN OPERATOR