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

2024/04/02 by Zhijie Chen, Chang‐Shou Lin, Chen, Zhijie +1
Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2404.01879

Abstract

The Darboux-Treibich-Verdier (DTV) potential ∑_k=0³n_k(n_k+1)\wp(z+\tfrac ω_k2;τ) is well-known as doubly-periodic solutions of the stationary KdV hierarchy (Treibich-Verdier, Duke Math. J. \bf 68 (1992), 217-236). In this paper, we study the generalized Lamé equation with the DTV potential y^{′ ′} (z)=[ ∑_k=0³n_k(n_k+1)\wp(z+\tfrac ω_k2;τ)+B] y(z), n_k∈ ℕ from the monodromy aspect. We prove that the map from (τ, B) to the monodromy data (r,s) satisfies a surprising universal law dτ\wedge dB≡8π² dr\wedge ds. Our proof applies Panlevé VI equation and modular forms. We also give applications to the algebraic multiplicity of (anti)periodic eigenvalues for the associated Hill operator.

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