Bethe Ansatz and Classical Hirota Equation

We discuss an interrelation between quantum integrable models and classical soliton equations with discretized time. It appeared that spectral characteristics of quantum integrable systems may be obtained from entirely classical set up. Namely, the eigenvalues of the quantum transfer matrix and the scattering $S$-matrix itself are identified with a certain $\tau$-functions of the discrete Liouville equation. The Bethe ansatz equations are obtained as dynamics of zeros. For comparison we also present the Bethe ansatz equations for elliptic solutions of the classical discrete Sine-Gordon equation. The paper is based on the recent study of classical integrable structures in quantum integrable systems, hep-th/9604080.Comment: 15 pages, Latex, special World Scientific macros include

On the singular spectrum of the Almost Mathieu operator. Arithmetics and Cantor spectra of integrable models

I review a recent progress towards solution of the Almost Mathieu equation (A.G. Abanov, J.C. Talstra, P.B. Wiegmann, Nucl. Phys. B 525, 571, 1998), known also as Harper's equation or Azbel-Hofstadter problem. The spectrum of this equation is known to be a pure singular continuum with a rich hierarchical structure. Few years ago it has been found that the almost Mathieu operator is integrable. An asymptotic solution of this operator became possible due analysis the Bethe Ansatz equations.Comment: Based on the lecture given at 13th Nishinomiya-Yukawa Memorial Symposium on Dynamics of Fields and Strings, Nishinomiya, Japan, 12-13 Nov 1998, and talk given at YITP Workshop on New Aspects of Strings and Fields, Kyoto, Japan, 16-18 Nov 199

Elliptic solutions to difference non-linear equations and related many-body problems

We study algebro-geometric (finite-gap) and elliptic solutions of fully discretized KP or 2D Toda equations. In bilinear form they are Hirota's difference equation for $\tau$-functions. Starting from a given algebraic curve, we express the $\tau$-function and the Baker-Akhiezer function in terms of the Riemann theta function. We show that the elliptic solutions, when the $\tau$-function is an elliptic polynomial, form a subclass of the general algebro-geometric solutions. We construct the algebraic curves of the elliptic solutions. The evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the Ruijsenaars-Schneider many body system. The zeros obey equations which have the form of nested Bethe-Ansatz equations, known from integrable quantum field theories. We discuss the Lax representation and the action-angle-type variables for the many body system. We also discuss elliptic solutions to discrete analogues of KdV, sine-Gordon and 1D Toda equations and describe the loci of the zeros.Comment: 22 pages, Latex with emlines2.st