  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 $\Delta=-1/2$ 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.
