Mathieu equation and Elliptic curve

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.

12 pages; minor improvement of the Conclusion section, references added

References in corpus (4)

• Deformed Prepotential, Quantum Integrable System and Liouville Field Theory
• Magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory
• TBA for the Toda chain
• Matone's relation of N=2 super Yang-Mills and spectrum of Toda chain

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• Non-perturbative studies of N=2 conformal quiver gauge theories
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• Exact partition functions for deformed $\mathcal{N}=2$ theories with $N_f=4$ flavours