Zeta-invariants of the Steklov spectrum for a planar domain

TL;DR

This paper introduces zeta-invariants derived from the Steklov spectrum of planar domains, which are conformally invariant and help in understanding inverse spectral problems.

Contribution

It defines new zeta-invariants based on Fourier coefficients that are uniquely determined by the Steklov spectrum and explores their properties and invariance.

Findings

Zeta-invariants are uniquely determined by the spectrum.

They are invariant under conformal transformations.

Open questions about the properties of zeta-invariants are posed.

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.