Mathieu equation and Elliptic curve

2010/06/30 by Wei He, Yan-Gang Miao · 1 citation
Physics and Astronomy · Mathematics · #math-ph #hep-th #math.MP

paper · pdf · doi:10.1088/0253-6102/58/6/08

published as Commun. Theor. Phys. 58(2012)827-834 · 12 pages; minor improvement of the Conclusion section, references added

arxiv created 2011/03/20 · arxiv updated 2015/05/19

Abstract

We present a relation between the Mathieu equation and a particular elliptic curve. We find that the Floquet exponent of the Mathieu equation, for both q<<1 and q>>1, can be obtained from the integral of a differential one form along the two homology cycles of the elliptic curve. Certain higher order differential operators are needed to generate the WKB expansion. We provide a fifth order proof.

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