New Fundamental Symmetries of Integrable Systems and Partial Bethe Ansatz

✍ Scribed by A.G. Ushveridze

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 596 KB
• Volume: 266
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1006/aphy.1998.5797

✦ Synopsis

We introduce a new concept of quasi-Yang Baxter algebras. These algebras, being simple but non-trivial deformations of ordinary algebras of monodromy matrices, realize a new type of quantum dynamical symmetry and find unexpected and remarkable applications in quantum inverse scattering method (QISM). We show that applying to quasi-Yang Baxter algebras the standard procedure of QISM one obtains new wide classes of quantum models which, being integrable (i.e., having enough number of commuting integrals of motion), are only quasi-exactly solvable (i.e., admit an algebraic Bethe ansatz solution for arbitrarily large but limited parts of the spectrum). These quasi-exactly solvable models naturally arise as deformations of known exactly solvable ones. A general theory of such deformations is proposed. The correspondence Yangian quasi-Yangian'' and XXX spin models quasi-XXX spin models'' is discussed in detail. We also construct the classical conterparts of quasi-Yang Baxter algebras and show that they naturally generate new classes of classical integrable models. We conjecture that these models are quasi-exactly solvable, i.e., admit only partial construction of action-angle variables in the framework of the classical inverse scattering method. The mathematical formalism elaborated in this paper naturally leads to the notions of quasi-commutators'' and quasi-Poisson brackets'' which can be considered as special deformations of the ordinary commutators and Poisson brackets and may play a fundamental role in the theory of integrable and quasi-exactly solvable systems.