TY - JOUR

T1 - Eight-vertex model and non-stationary Lamé equation

AU - Bazhanov, Vladimir V.

AU - Mangazeev, Vladimir V.

PY - 2005/2/25

Y1 - 2005/2/25

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=14544278909&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/38/8/L01

DO - 10.1088/0305-4470/38/8/L01

M3 - Article

SN - 0305-4470

VL - 38

SP - L145-L153

JO - Journal of Physics A: Mathematical and General

JF - Journal of Physics A: Mathematical and General

IS - 8

ER -