Singular Finite-Gap Operators and Indefinite Metric
Singular Finite-Gap Operators and Indefinite Metric
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奇异有限间隙算子和不定度量
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
U. Park
中科院分区：
文献类型：
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作者：
P. Grinevich;S. L. F. P. Physics;U. Park
周期性有限间隙算子的许多“真实”反谱数据（由带有标记“无限点”的黎曼曲面、局部参数和极点除数组成）导致算子具有真实但奇异的系数。这些运算符不能被视为在 x 函数的普通（正）希尔伯特空间中自伴。特别是，对于具有椭圆势 $n(n+1)wp(x)$ 的 Lame 算子的特殊情况来说，这是正确的，其中特征函数是由 Hermit 在 19 世纪发现的。然而，这种 Baker-Akhiezer (BA) 函数是根据 Krichever-Novikov (1989)、Grinevich-Novikov (2001) 的著作思想提出的，它是黎曼曲面上的离散和连续傅里叶基的正确模拟。事实证明，这些非零属的运算符在某些不定内积中是对称的，如本工作中所述。连续傅里叶变换的模拟是该内积的等距。在下一个工作中，我们将阐述离散傅立叶级数的类似理论
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