International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 9, Pages 513-553

Quantum relativistic Toda chain at root of unity: isospectrality, modified Q-operator, and functional Bethe ansatz

Stanislav Pakuliak and Sergei Sergeev

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.


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.