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New separation of variables for the classical \(XXX\) and \(XXZ\) Heisenberg spin chains. (English) Zbl 1444.37049

Summary: We propose a non-standard separation of variables for the classical integrable XXX and XXZ spin chains with degenerate twist matrix. We show that for the case of such twist matrices one can interchange the role of classical separating functions \(A(u)\) and \(B(u)\) and construct a new full set of separated variables, satisfying simpler equation of separation and simpler Abel equations in comparison with the standard separated variables of Sklyanin. We show that for certain cases of the twist matrices the constructed separated variables can be directly identified with action-angle coordinates.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37J37 Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures
17B80 Applications of Lie algebras and superalgebras to integrable systems
82B23 Exactly solvable models; Bethe ansatz
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