Eight-vertex model and non-stationary Lame 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 $\mathrm{\Delta }=-1\mathrm{/}2$ six-vertex model. We show that these eigenvalues satisfy a non-stationary Schrodinger equation with the time-dependent potential given by the Weierstrass elliptic P-function where the modular parameter $\tau$ 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 Painleve III equation.