Separation of variables, Lax-integrable systems and \(gl(2) \otimes gl(2)\)-valued classical \(r\)-matrices
This paper discusses a separation of variables problem for a certain class of Lax-integrable Hamiltonian systems. It addresses specific questions about the construction of separated variables in the context of integrable Hamiltonian systems. The separated variables \(x_i\) and \(p_j\) (\(i, j =1, \dots, d\)) are a set of canonical coordinates that satisfy a general system of equations for separation. That is, \(\Phi_i (x_i, p_i, I_1, ..., I_d, C_1, ..., C_r) = 0\) for \(i = 1, \dots, d\). The \(I_k\)'s are Poisson-commuting integrals of motion, the \(C_k\)'s are Casimir functions, and \(d\) is one-half the dimension of the symplectic leaves in the phase space. Separated variables are typically used to allow explicit integration of the equations of motion. Separation of variables is therefore of interest for integrable systems in both classical and quantum physics. The author is specifically interested in integrable Hamiltonian systems for which a Lax representation with spectral parameter is known. He calls Lax-integrable systems those systems for which the spectral invariants of the Lax matrix comprise a complete set of Poisson-commuting integrals of motion. A conjecture called ``magic recipe'', that goes back to \textit{E. K. Sklyanin} [J. Sov. Math. 47, No. 2, 2473--2488 (1989; Zbl 0692.35107); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 164, 151--169 (1987)], is recalled here. An open question about this ``magic recipe'' is when the variables constructed in this way are actually canonical. This was answered by \textit{B. Dubrovin} and \textit{T. Skrypnyk} [J. Math. Phys. 59, No. 9, 091405, 39 p. (2018; Zbl 1403.37067)], who described general separability criteria. In the current paper the author considers what is necessary to produce a set of canonical variables that is complete in the simplest non-trivial case. It would be advisable for a reader interested in this paper to review first at least the two references cited above. The current paper provides a somewhat sketchy background and does not completely define some of the terms it uses.
scientific article; zbMATH DE number 7223539
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