Monodromy of generalized Lamé equations with Darboux-Treibich-Verdier potentials: A universal law

Abstract

The Darboux-Treibich-Verdier (DTV) potential ${\sum }_{k=0}^3{n}_k(n_k+1)\wp (z+\frac{{\omega }_k}{2};\tau )$ is well-known as a (parametric) doubly-periodic solution of the stationary KdV hierarchy (Treibich and Verdier, 1992) [42]. In this paper, we study the generalized Lamé equation with the DTV potential$y^{″}\left(z\right)=\left[\sum _{k=0}^3{n}_k\right(n_k+1)\wp (z+\frac{{\omega }_k}{2};\tau )+B]y\left(z\right),n_k\in N$ from the (doubly periodic) monodromy aspect. This equation is the elliptic form of the well-known integral Heun equation. We prove that the map from $(\tau ,B)$ to the monodromy data $(r,s)$ satisfies a surprising universal law $d\tau \wedge dB\equiv 8{\pi }^{2}dr\wedge ds$. Our proof applies the Panlevé VI equation and modular forms. We also give applications to compute the algebraic multiplicity of the (anti)periodic eigenvalues for the associated Hill operator.