平面域Steklov谱的Zeta不变量
Zeta-invariants of the Steklov spectrum for a planar domain
摘要：
从Steklov谱\cite{E}恢复单连通光滑平面域的经典反问题等价于从算子$aΛ_e$的特征谱恢复单位圆${\mathbb S}=\{e^{iθ}\}$上的正函数$a\in C^\infty({\mathbb S})$直到保形等价的问题，其中$Λ_e=(-d^2/dθ^2)^{1/2}$。我们在函数$a$的傅里叶系数中引入了$2k$ -形式$Z_k(a)\ (k=1,2,\dots)$，称为ζ-不变量。它们由$aΛ_e$的本征值谱唯一确定。研究了$Z_k(a)$的一些性质，特别是在共形群下的不变性.在论文的最后提出了一些关于zeta不变量的开放性问题。
英文摘要：
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.
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