International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 9, Pages 513-553
doi:10.1155/S0161171202105059
Quantum relativistic Toda chain at root of
unity: isospectrality, modified Q-operator, and functional Bethe ansatz
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Moscow, Russia
Received 17 May 2001; Revised 9 April 2002
Copyright © 2002 Stanislav Pakuliak and Sergei Sergeev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We investigate an N-state spin model called quantum
relativistic Toda chain and based on the unitary
finite-dimensional representations of the Weyl algebra with q
being Nth primitive root of unity. Parameters of the
finite-dimensional representation of the local Weyl algebra form
the classical discrete integrable system. Nontrivial dynamics of
the classical counterpart corresponds to isospectral
transformations of the spin system. Similarity operators are
constructed with the help of modified Baxter's Q-operators. The
classical counterpart of the modified Q-operator for the initial
homogeneous spin chain is a Bäcklund transformation.
This transformation creates an extra Hirota-type soliton in a
parameterization of the chain structure. Special choice of values
of solitonic amplitudes yields a degeneration of spin
eigenstates, leading to the quantum separation of variables, or
the functional Bethe ansatz. A projector to the separated
eigenstates is constructed explicitly as a product of modified
Q-operators.