从薛定谔谱到正交多项式，通过一个函数方程
From Schrödinger spectra to orthogonal polynomials, via a functional equation
摘要：
线性薛定谔算子的某些谱问题，例如几乎马蒂厄方程，与正交多项式的三项递推关系之间的主要区别在于，在前者中，指数范围跨越$\ZZ$，而在后者中，指数范围仅跨越$\Zp$。我们提出了一种技术，通过Dirichlet和Taylor展开的混合，几乎马蒂厄方程及其推广到三项递归关系。这开辟了充分利用正交多项式理论的力量在薛定谔光谱分析的可能性。上述三项递推关系式都具有系数几乎周期的性质。我们推广的证明方法，由于最初的杰夫Geronimo和Walter van Assche，调查相关的正交多项式的Borel测度的基本支持，从而推导出薛定谔算子的基本绝对连续谱的信息。
英文摘要：
The main difference between certain spectral problems for linear Schrödinger operators, e.g. the almost Mathieu equation, and three-term recurrence relations for orthogonal polynomials is that in the former the index ranges across $\ZZ$ and in the latter only across $\Zp$. We present a technique that, by a mixture of Dirichlet and Taylor expansions, translates the almost Mathieu equation and its generalizations to three term recurrence relations. This opens up the possibility of exploiting the full power of the theory of orthogonal polynomials in the analysis of Schrödinger spectra. Aforementioned three-term recurrence relations share the property that their coefficients are almost periodic. We generalize a method of proof, due originally to Jeff Geronimo and Walter van Assche, to investigate essential support of the Borel measure of associated orthogonal polynomials, thereby deriving information on the underlying absolutely continuous spectra of Schrödinger operators.
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