Eight-vertex model and non-stationary Lamé equation

We study the ground state eigenvalues of Baxter's Q-operator for the eight-vertex model in a special case when it describes the off-critical deformation of the Δ = -1/2 six-vertex model. We show that these eigenvalues satisfy a non-stationary Schrödinger equation with the time-dependent potential given by the Weierstrass elliptic ℘-function where the modular parameter τ plays the role of (imaginary) time. In the scaling limit, the equation transforms into a 'non-stationary Mathieu equation' for the vacuum eigenvalues of the Q-operators in the finite-volume massive sine-Gordon model at the super-symmetric point, which is closely related to the theory of dilute polymers on a cylinder and the Painlevé III equation.