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 $\mathbb{S}=\{e^{i\theta }\}$ from the eigenvalue spectrum of the operator $a{\Lambda }_e$, where ${\Lambda }_e=(-d^2/d{\theta }^2{)}^{1/2}$. 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 $a{\Lambda }_e$. 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.