The two-dimensional quantum lattice Toda model for the affine and simple Lie algebras of the type A is considered. For its known L-operator a correction of the second order in the lattice parameter is found. It is proved that the equation determining a correction of the third order has no solutions.
Quantum Hamiltonian systems of lattice type are considered which are associated with classical finite-dimensional Lie algebras [like MH(2), SO(3), etc.], and possess complete integrability properties. The canonical Heisenberg pn, qn variables are found for these Hamiltonian systems and it is shown how to construct their S matrices. These new Hamiltonians are generalizations of the quantum Toda lattice Hamiltonian as well as XXX models of statistical mechanics.
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 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.
New boundary conditions for classical integrable nonlinear lattices of the XXX type, such as the Heisenberg chain and the Toda lattice, are presented. These integrable extensions are formulated in terms of a generic XXX Heisenberg magnet interacting with two additional spins at each end of the chain. The construction uses the most general rank-1 ansatz for the 2*2 L-operator satisfying the reflection equation algebra with rational r-matrix. The associated quadratic algebra is shown to be that of dynamical symmetry for the A 1 and BC 2 Calogero-Moser problems. Other physical realizations of our quadratic algebra are also considered.
We consider solvable lattice models D L +2 (1) (λ) associated with the Dynkin diagrams of the affine Lie algebras D L +2 (1) and parametrized by the crossing parameter λ (0 < λ < 2π/3). The transfer matrix spectra for the strip widths ⩽ 10 are classified according to the irreducible representations of the symmetry groups under several boundary conditions. All L - and λ-dependences of scaling dimensions are embodied in the thermal scaling dimension X T = 2/{ L 2 (1 - λ/π)}. The operator content of the model is the same as that of the Ashkin-Teller model with the same thermal scaling dimension. This generalizes the results of Pasquier (λ = 0) and of the magnetic hard square model ( L = 3).
A lattice version of the quantum nonlinear Schrodinger (NLS) equation is considered, which has a significant simple form and fulfils most of the criteria desirable for such lattice variants of field models. Unlike most of the known lattice NLS equations, the present model belongs to a class which does not exhibit the usual symmetry properties. However, this lack of symmetry itself seems to be responsible for the remarkable simplification of the relevant objects in the theory, such as the Lax operator, the Hamiltonian and other commuting conserved quantities as well as their spectra. The model allows exact quantum solution through the algebraic Bethe ansatz and also a straightforward and natural generalization to the vector case, thus giving a new exact lattice version of the vector NLS model. A deformation representing a new quantum integrable system involving Tamm-Dancoff-like q-boson operators is constructed.