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**Zeta-invariants of the Steklov spectrum of a planar domain.**
*(English.
Russian original)*
Zbl 1328.35319

Sib. Math. J. 56, No. 4, 678-698 (2015); translation from Sib. Mat. Zh. 56, No. 4, 853-877 (2015).

The Steklov spectrum of a domain \(\Omega\subset\mathbb{R}^2\) consists of those \(\lambda\in\mathbb{R}\) for which the following boundary problem has a non-trivial solution: \(\Delta u=0\) in \(\Omega\) and \(\partial u/\partial\nu +\lambda u=0\) on the boundary. The authors assume that \(\Omega\) is simply-connected and bounded by a smooth closed curve. They ask to which extent \(\Omega\) is determined by its Steklov spectrum. This is an inverse problem for the associated Dirichlet-to-Neumann (DN) operator. Moreover, using a conformal mapping \(\Phi\) from the unit disk \(\mathbb{D}\) onto \(\Omega\), the problem is turned into an inverse spectral problem for a first-order pseudodifferential operator \(a\Lambda_e\) on the unit circle \(\mathbb{S}\). Here, \(0<a\in C^\infty(\mathbb{S})\) is the reciprocal of \(|\Phi'|\) restricted to \(\mathbb{S}\) and \(\Lambda_e=(-d^2/d\theta^2)^{1/2}\) is the DN operator of the Euclidean metric on \(\mathbb{D}\).

Given \(a\in C^\infty(\mathbb{S})\) and an integer \(k\geq 1\), the authors define zeta-invariants \(Z_k(a)\in\mathbb{C}\) as special \(2k\)-linear forms in the Fourier coefficients of \(a\). Generalizing a result of J. Edward [J. Funct. Anal. 111, No. 2, 312–322 (1993; Zbl 0813.47003)] from \(k=1\) to \(k\geq 1\), the authors prove that if \(a>0\) and normalized, then \(Z_k(a)=\zeta_a(-2k)\). Here, the zeta-function \(\zeta_a(s)\) is the trace of \((a\Lambda_e)^{-s}\). It follows that the isospectrality of \(a\Lambda_e\) and \(b\Lambda_e\) implies the equality of all zeta-invariants, \(Z_a(k)=Z_b(k)\). Therefore, the authors pose the completeness problem of zeta-invariants: Given \(b>0\) smooth, find all positive smooth functions \(a\) having the same zeta-invariants as \(b\). The authors admit that they are far from a final solution of the problem. However, in the second half of the paper, they derive some invariance properties of the zeta-invariants. Moreover, they determine \(Z_2\) explicitly.

Given \(a\in C^\infty(\mathbb{S})\) and an integer \(k\geq 1\), the authors define zeta-invariants \(Z_k(a)\in\mathbb{C}\) as special \(2k\)-linear forms in the Fourier coefficients of \(a\). Generalizing a result of J. Edward [J. Funct. Anal. 111, No. 2, 312–322 (1993; Zbl 0813.47003)] from \(k=1\) to \(k\geq 1\), the authors prove that if \(a>0\) and normalized, then \(Z_k(a)=\zeta_a(-2k)\). Here, the zeta-function \(\zeta_a(s)\) is the trace of \((a\Lambda_e)^{-s}\). It follows that the isospectrality of \(a\Lambda_e\) and \(b\Lambda_e\) implies the equality of all zeta-invariants, \(Z_a(k)=Z_b(k)\). Therefore, the authors pose the completeness problem of zeta-invariants: Given \(b>0\) smooth, find all positive smooth functions \(a\) having the same zeta-invariants as \(b\). The authors admit that they are far from a final solution of the problem. However, in the second half of the paper, they derive some invariance properties of the zeta-invariants. Moreover, they determine \(Z_2\) explicitly.

Reviewer: Sönke Hansen (Paderborn)

### MSC:

35R30 | Inverse problems for PDEs |

35P20 | Asymptotic distributions of eigenvalues in context of PDEs |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

58J53 | Isospectrality |

### Citations:

Zbl 0813.47003
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\textit{E. G. Mal'kovich} and \textit{V. A. Sharafutdinov}, Sib. Math. J. 56, No. 4, 678--698 (2015; Zbl 1328.35319); translation from Sib. Mat. Zh. 56, No. 4, 853--877 (2015)

### References:

[1] | Edward, J., “an inverse spectral result for the Neumann operator on planar domains,”, J. Funct. Anal., 111, 312-322, (1993) · Zbl 0813.47003 |

[2] | Jollivet, A.; Sharafutdinov, V., “on an inverse problem for the Steklov spectrum of a Riemannian surface,”, Contemp. Math., 615, 165-191, (2014) · Zbl 1338.58017 |

[3] | Prudnikov A. P., Brychkov Yu. A., and Marichev O. I., Integrals and Series [in Russian], Nauka, Moscow (1981). · Zbl 0511.00044 |

[4] | Brooks, R.; Perry, P.; Petersen, P., “compactness and finiteness theorems for isospectral manifolds,”, J. Reine Angew. Math., 426, 67-89, (1992) · Zbl 0737.53038 |

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