Monodromy of generalized Lamé equations with Darboux-Treibich-Verdier potentials: A universal law
Journal
Advances in Mathematics
Journal Volume
468
Start Page
110209
ISSN
0001-8708
Date Issued
2025-05
Author(s)
Zhijie Chen
Abstract
The Darboux-Treibich-Verdier (DTV) potential [Formula presented] 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 [Formula presented] 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 (τ,B) to the monodromy data (r,s) satisfies a surprising universal law dτ∧dB≡8π2dr∧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.
SDGs
Publisher
Elsevier BV
Type
journal article