The quantum open Toda lattice is studied by the method of the separation of variable developed by Sklyanin. The quantum canonical operators are derived from the monodromy matrix. Discussed is also the relativistic open Toda lattice.
We consider quantum analogs of the relativistic Toda lattices and give new 2×2L-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 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.
We reconstruct the canonical operators pi,qi of the quantum closed Toda chain in terms of Sklyanin's separated variables.
We consider quantum analogs of the relativistic Toda lattices and give new 2×2L-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 present a microscopic approach in the framework of Sklyanin's quantum separation of variables (SOV) for the exact solution of 1+1-dimensional quantum field theories by integrable lattice regularizations. Sklyanin's SOV is the natural quantum analogue of the classical method of separation of variables and it allows a more symmetric description of classical and quantum integrability w.r.t. traditional Bethe ansatz methods. Moreover, it has the advantage to be applicable to a more general class of models for which its implementation gives a characterization of the spectrum complete by construction. Our aim is to introduce a method in this framework which allows at once to derive the spectrum (eigenvalues and eigenvectors) and the dynamics (time dependent correlation functions) of integrable quantum field theories (IQFTs). This approach is presented for a paradigmatic example of relativistic IQFT, the sine-Gordon model.