Lamé多项式、超椭圆约化和Lamé能带结构
Lamé polynomials, hyperelliptic reductions and Lamé band structure
摘要：
研究了一维周期势薛定谔方程的Lamé方程的能带结构.在整数值的程度参数l，色散关系减少到l=1的色散关系，和先前公布的l=2的色散关系被证明是部分不正确的。Hermite-Krichever Answer是主要的工具，它将Lamé方程的解表示为l=1的解。它是基于从一个genus-l超椭圆曲线，参数化的解决方案，椭圆曲线的投影。这个覆盖的一般公式推导出来，并用于减少某些超椭圆积分椭圆的。带边之间的退化，这可能会发生，如果Lamé方程参数采取复杂的值，进行了研究。如果把Lamé方程看作是椭圆曲线上的微分方程，则给出了椭圆模空间（椭圆曲线参数空间）中发生退化的点的个数的公式。谱多项式和Lamé多项式表，即，带边解。更正了旧文献中的一个表格。
英文摘要：
The band structure of the Lamé equation, viewed as a one-dimensional Schrödinger equation with a periodic potential, is studied. At integer values of the degree parameter l, the dispersion relation is reduced to the l=1 dispersion relation, and a previously published l=2 dispersion relation is shown to be partially incorrect. The Hermite-Krichever Ansatz, which expresses Lamé equation solutions in terms of l=1 solutions, is the chief tool. It is based on a projection from a genus-l hyperelliptic curve, which parametrizes solutions, to an elliptic curve. A general formula for this covering is derived, and is used to reduce certain hyperelliptic integrals to elliptic ones. Degeneracies between band edges, which can occur if the Lamé equation parameters take complex values, are investigated. If the Lamé equation is viewed as a differential equation on an elliptic curve, a formula is conjectured for the number of points in elliptic moduli space (elliptic curve parameter space) at which degeneracies occur. Tables of spectral polynomials and Lamé polynomials, i.e., band edge solutions, are given. A table in the older literature is corrected.
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