Zeta-invariants of the Steklov spectrum of a planar domain

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 $\mathbb S=\{e^{i\theta}\}$ from the spectrum of the operator $a\Lambda_e$, where $\Lambda_e=(-d^2/d\theta^2)^{1/2}$. We introduce $2k$-forms $Z_k(a)$ ($k=1,2,\dots$) of the Fourier coefficients of $a$, called the zeta-invariants. These invariants are determined by the eigenvalues of $a\Lambda_e$. 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.