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Separation of variables for the quantum relativistic Toda lattices

Vadim B. Kuznetsov, Andrey Vladimirovich Tsiganov

Published Feb 18, 1994• 0 citations
OpenAlex

Abstract

We consider quantum analogs of the relativistic Toda lattices and give new $2\times 2$ $L$-operators for these models. Making use of the variable separation the spectral problem for the quantum integrals of motion is reduced to solving one-dimensional separation equations.

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We consider quantum analogs of the relativistic Toda lattices and give new 2×22\times 22×2 LLL-operators for these models. Making use of the variable separation the spectral problem for the quantum integrals of motion is reduced to solving one-dimensional separation equations.
We consider quantum analogs of the relativistic Toda lattices and give new 2×22\times 22×2 LLL-operators for these models. Making use of the variable separation the spectral problem for the quantum integrals of motion is reduced to solving one-dimensional separation equations.
A new integrable class of quantum models representing a family of different discrete-time or relativistic generalisations of the periodic Toda chain (TC), including that of a recently proposed classical close to TC model [7] is presented. All such models are shown to be obtainable from a single ancestor model at different realisations of the underlying quantised algebra. As a consequence the (2times2)(2\\times 2)(2times2) Lax operators and the associated quantum RRR-matrices for these models are easily derived ensuring their quantum integrability. It is shown that the functional Bethe ansatz developed for the quntum TC is trivially generalised to achieve seperation of variables also for the present models.
We generalize Babelon's approach to equations in dual variables so as to be able to treat new types of operators which we build out of the sub-constituents of the model's monodromy matrix. Further, we also apply Sklyanin's recent monodromy matrix identities so as to obtain equations in dual variables for yet other operators. The schemes discussed in this paper appear to be universal and thus, in principle, applicable to many models solvable through the quantum separation of variables.