949 research outputs found
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 -matrix itself are identified with a certain -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 -functions. Starting from a given algebraic
curve, we express the -function and the Baker-Akhiezer function in terms
of the Riemann theta function. We show that the elliptic solutions, when the
-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
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