Zeta-invariants of the Steklov spectrum of a planar domain

Abstract

The classical inverse problem of the determination of a smooth simply-connected planar domain by its Steklov spectrum [1] is equivalent to the problem of the reconstruction, up to conformal equivalence, a positive function $a \in C^\infty (\mathbb{S})$ on the unit circle $(\mathbb{S}) = \left\{ {e^{i\theta } } \right\}$ from the spectrum of the operator aΛ _e , where Λ _e = (−d ²/dθ ²)^1/2. We introduce 2k-forms Z _k (a) (k = 1, 2,... ) of the Fourier coefficients of a, called the zeta-invariants. These invariants are determined by the eigenvalues of aΛ _e . We study some properties of the forms Z _k (a); in particular, their invariance under the conformal group. A few open questions about zeta-invariants is posed at the end of the article.