Zeta-invariants of the Steklov spectrum for a planar domain

Abstract

The classical inverse problem of recovering a simply connected smooth planar domain from the Steklov spectrum \cite{E} is equivalent to the problem of recovering, up to a conformal equivalence, a positive function $a\in C^\infty({\mathbb S})$ on the unit circle ${\mathbb S}=\{e^{iθ}\}$ from the eigenvalue spectrum of the operator $aΛ_e$, where $Λ_e=(-d^2/dθ^2)^{1/2}$. We introduce $2k$-forms $Z_k(a)\ (k=1,2,\dots)$ in Fourier coefficients of the function $a$ which are called zeta-invariants. They are uniquely determined by the eigenvalue spectrum of $aΛ_e$. We study some properties of $Z_k(a)$, in particular, their invariance under the conformal group. Some open questions on zeta-invariants are posed at the end of the paper.