This article is cited in 2 papers

Quantum Bäcklund transformations: Some ideas and examples

O. Ragnisco^ab, F. Zullo<sup>c

a</sup> Istituto Nazionale di Fisica Nucleare, sezione di Roma Tre, Rome, Italy
^b Dipartimento di Fisica, Università di Roma Tre, Rome, Italy
^c School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, Kent, U.K.

Abstract: We propose a mechanical (Hamiltonian) interpretation of the so-called spectrality property introduced by Sklyanin and Kuznetsov in the context of Bäcklund transformations (BTs) for finite-dimensional integrable systems. This property turns out to be deeply connected with the Hamilton–Jacobi separation of variables and can lead to the explicit integration of the corresponding model using the BTs. We show that once such a construction is given, we can interpret the Baxter $Q$-operator defining the quantum BTs as the Green's function or the propagator of the time-dependent Schrödinger equation for an interpolating Hamiltonian.

Keywords: quantum Bäcklund transformation, spectrality property, integrable map, quantum propagator.

DOI: 10.4213/tmf6950

 English version: Theoretical and Mathematical Physics, 2012, 172:2, 1160–1171

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