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Quantum relativistic Toda chain. (Russian. English summary) Zbl 0821.17026

The relativistic Toda chain, which is a generalization of a classical periodic Toda chain, is studied. A description of the model in terms of Lax triads is given. This gives a possibility to prove complete integrability and define local integrals of motion. On the other hand, an inverse scattering method is developed, and a quantum analog of the system with standard trigonometric \(R\)-matrix is defined. Sklyanin’s program of separation of variables is realized both in the classical and quantum cases. The corresponding spectral problem is written down explicitly. All the results are generalized to the case of integrable boundary conditions of \(B\), \(C\), \(D\) types.

MSC:

17B80 Applications of Lie algebras and superalgebras to integrable systems
82B23 Exactly solvable models; Bethe ansatz
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
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