Quantum Toda chains intertwined. (English. Russian original) Zbl 1219.81128

Gerasimov, A.; Lebedev, D.; Oblezin, S.

St. Petersbg. Math. J. 22, No. 3, 411-435 (2011); translation from Algebra Anal. 2010, No. 3, 107-141 (2010).

Summary: An explicit construction of integral operators intertwining various quantum Toda chains is conjectured. Compositions of the intertwining operators provide recursive and \( \mathcal{Q}\)-operators for quantum Toda chains. In particular the authors’ earlier results on Toda chains corresponding to classical Lie algebras are extended to the generic \( BC_n\)- and Inozemtsev-Toda chains. Also, an explicit form of \( \mathcal{Q}\)-operators is conjectured for the closed Toda chains corresponding to the Lie algebras \( B_{\infty}, C_{\infty}, D_{\infty}\), the affine Lie algebras \( B^{(1)}_n, C^{(1)}_n, D^{(1)}_n, D^{(2)}_n, A^{(2)}_{2n-1}, A^{(2)}_{2n}\), and the affine analogs of \( BC_n\)- and Inozemtsev-Toda chains.

MSC:

81Q12	Nonselfadjoint operator theory in quantum theory including creation and destruction operators
81S05	Commutation relations and statistics as related to quantum mechanics (general)
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Keywords:

quantum Toda Hamiltonians; elementary intertwining operator; recursive operator; quantization Pasquier-Gaudin integral \(Q\)-operator

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