奇异Chronte-Gap算子与不定度量
Singular Finite-Gap Operators and Indefinite Metric
摘要：
周期有限间隙算子的许多“实”逆谱数据（由具有标记“无穷远点”的Riemann曲面、局部参数和极点因子组成）导致具有实但奇异系数的算子。这些算子在x的函数的普通（正）希尔伯特空间中不能被认为是自伴的。特别是，这是真实的特殊情况下，拉梅运营商与椭圆型潜在的$n(n+1)\wp(x)$，本征函数发现在十九世纪的埃尔米特。然而，根据Krichever-Novikov（1989），Grinevich-Novikov（2001）的作品的思想，这样的Baker-Akhiezer（BA）函数存在Riemann曲面上的离散和连续傅立叶基的右模拟。事实证明，这些运营商的非零亏格是对称的一些不确定的内积，在这项工作中描述。连续傅里叶变换的模拟是这个内积的等距。在下一个工作与第二，我们将提出类似的离散傅立叶级数理论的阐述
英文摘要：
Many "real" inverse spectral data for periodic finite-gap operators (consisting of Riemann Surface with marked "infinite point", local parameter and divisors of poles) lead to operators with real but singular coefficients. These operators cannot be considered as self-adjoint in the ordinary (positive) Hilbert spaces of functions of x. In particular, it is true for the special case of Lame operators with elliptic potential $n(n+1)\wp(x)$ where eigenfunctions were found in XIX Century by Hermit. However, such Baker-Akhiezer (BA) functions present according to the ideas of works by Krichever-Novikov (1989), Grinevich-Novikov (2001) right analog of the Discrete and Continuous Fourier Bases on Riemann Surfaces. It turns out that these operators for the nonzero genus are symmetric in some indefinite inner product, described in this work. The analog of Continuous Fourier Transform is an isometry in this inner product. In the next work with number II we will present exposition of the similar theory for Discrete Fourier Series
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