A new kink method computes the Floquet phase of Heun-type equations as convergent series, matching the dual gauge period of N=2 SYM in the NS background and giving black-hole perturbation wavefunctions.
Mathieu equation and Elliptic curve
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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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Regular and Floquet bases for gauge and gravity theories: a non perturbative approach
A new kink method computes the Floquet phase of Heun-type equations as convergent series, matching the dual gauge period of N=2 SYM in the NS background and giving black-hole perturbation wavefunctions.