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.


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


Zbl 0813.47003
Full Text: DOI arXiv


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.