Open Toda Chain Research Articles

Equations in Dual Variables for Whittaker Functions

It is known that the Whittaker functions w(qλ) associated with the group GL(N) are eigenfunctions of the Hamiltonians of the open Toda chain, hence satisfy a set of differential equations in the Toda variables q i . Using the expression of the q i for the closed Toda chain in terms of Sklyanin variables λ i , and the known relations between the open and the closed Toda chains, we show that Whittaker functions also satisfy a set of new difference equations in λ i .

Unitary Representations of U q (𝔰𝔩}(2,ℝ)),¶the Modular Double and the Multiparticle q -Deformed¶Toda Chain

The paper deals with the analytic theory of the quantum q-deformed Toda chain; the technique used combines the methods of representation theory and the Quantum Inverse Scattering Method. The key phenomenon which is under scrutiny is the role of the modular duality concept (first discovered by L.Faddeev) in the representation theory of noncompact semisimple quantum groups. Explicit formulae for the Whittaker vectors are presented in terms of the double sine functions and the wave functions of the N-particle q-deformed open Toda chain are given as a multiple integral of the Mellin-Barnes type. For the periodic chain the two dual Baxter equations are derived.

Unitary Representations of the Quantum Lorentz Group and Quantum Relativistic Toda Chain

We give a group theory interpretation of the three types of q-Bessel functions. We consider a family of quantum Lorentz groups and a family of quantum Lobachevsky spaces. For three values of the parameter of the quantum Lobachevsky space, the Casimir operators correspond to the two-body relativistic open Toda-chain Hamiltonians whose eigenfunctions are the modified q-Bessel functions of the three types. We construct the principal series of unitary irreducible representations of the quantum Lorentz groups. Special matrix elements in the irreducible spaces given by the q-Macdonald functions are the wave functions of the two-body relativistic open Toda chain. We obtain integral representations for these functions.

Integral representations for the eigenfunctions of quantum open and periodic Toda chains from the QISM formalism

The integral representations for the eigenfunctions of N particle quantum open and periodic Toda chains are constructed within the framework of the quantum inverse scattering method. Both periodic and open N-particle solutions have essentially the same structure, being written as a generalized Fourier transform over the eigenfunctions of the N-1 particle open Toda chain with the kernels satisfying the Baxter equations of second and first order, respectively. In the latter case this leads to recurrent relations which result in a representation of Mellin-Burnes-type solutions of an open chain. As a byproduct, we obtain the Gindikin-Karpelevich formula for the Harish-Chandra function in the case of the {GL}(N,) group.

D-dimensionalp-brane cosmological models associated with a Lie algebra of the typeA m

We study a D-dimensional cosmological model on the manifold M=R×M0×R×Mn describing an evolution of n+1 Einstein factor spaces Mi in a theory with several dilatonic scalar fields and differential forms admitting an interpretation in terms of intersecting p-branes. The equations of motion of the model are reduced to the Euler-Lagrange equations for the so-called pseudo-Euclidean Toda-like system. Assuming that the characteristic vectors related to the configuration of p-branes and their couplings to the dilatonic scalar fields can be interpreted as the root vectors of a Lie algebra of the type Am≡sl(m+1C), we reduce the model to an open Toda chain, which is integrable by the customary methods. The resulting metric has the form of the Kasner solution. We single out the particular model describing the Friedmanlike evolution of the three-dimensional external factor space M0 (in the Einsteinian conformal gauge) and the contraction of the internal factor spaces M1,...,Mn.

Eigenfunctions of GL(N, ℝ) Toda chain: Mellin-Barnes representation

The recurrent relations between the eigenfunctions for $GL(N,\RR)$ and $GL(N-1,\RR)$ quantum Toda chains is derived. As a corollary, the Mellin-Barnes integral representation for the eigenfunctions of a quantum open Toda chain is constructed for the $N$-particle case.

Toda Chains with Type Am Lie Algebra for Multidimensional m-component Perfect Fluid Cosmology

We consider a D-dimensional cosmological model describing an evolution of Ricci-flat factor spaces, M_1,...M_n (n > 2), in the presence of an m-component perfect fluid source (n > m > 1). We find characteristic vectors, related to the matter constants in the barotropic equations of state for fluid components of all factor spaces. We show that, in the case where we can interpret these vectors as the root vectors of a Lie algebra of Cartan type A_m=sl(m+1,C), the model reduces to the classical open m-body Toda chain. Using an elegant technique by Anderson (J. Math. Phys. 37 (1996) 1349) for solving this system, we integrate the Einstein equations for the model and present the metric in a Kasner-like form.

The quantum mechanical Toda lattice, II

The periodic Toda lattice consists of N particles which move along a closed line and are coupled with an exponential spring to their immediate neighbors. This system, in contrast to the open Toda lattice, has only bound states. In the method of Kac and Van Moerbeke, the classical periodic Toda chain is transformed to a new of set of canonically conjugate variables, μ and ν, which are closely related to the natural coordinates of an open Toda chain with one particle less. The quantum mechanical eigenfunctions for this reduced system are constructed explicitly, and this allows the quantum mechanical analogs of μ and ν to be defined. The bounds states for the periodic Toda chain are then written as linear combinations of functions resembling the wave functions of the reduced open chain. These functions satisfy a set of remarkably simple recursion formulas, and the coefficients in the expansion can be written essentially as a product of as many factors as pairs of conjugate variables μ and ν. Each factor is given as a solution of a second order difference equation which can be recognized as a quantum analog for the equations of motion of one pair μ and ν. The quantization conditions result from cancelling out the exponential growth in the overall wave function, and are phrased in terms of certain phase angles being submultiples of π according as the representation of the group of cyclic permutations. The calculations are simple for N = 3, and moderately tricky for N = 4 although the results are always fairly obvious.