## 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.

### 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

Zbl 0813.47003
Full Text:

### References:

  Edward, J., “an inverse spectral result for the Neumann operator on planar domains,”, J. Funct. Anal., 111, 312-322, (1993) · Zbl 0813.47003  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  Prudnikov A. P., Brychkov Yu. A., and Marichev O. I., Integrals and Series [in Russian], Nauka, Moscow (1981). · Zbl 0511.00044  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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