Affine Toda–Sutherland systems

Abstract

A cross between two well-known integrable multi-particle dynamics, an affine Toda molecule and a Sutherland system, is introduced for any affine root system. Though it is not completely integrable but partially integrable, or quasi exactly solvable, it inherits many remarkable properties from the parents. The equilibrium position is algebraic, i.e. proportional to the Weyl vector. The frequencies of small oscillations near equilibrium are proportional to the affine Toda masses, which are essential ingredients of the exact factorisable S-matrices of affine Toda field theories. Some lower lying frequencies are integer times a coupling constant for which the corresponding exact quantum eigenvalues and eigenfunctions are obtained. An affine Toda-Calogero system, with a corresponding rational potential, is also discussed.

A novel quasi-exactly solvable spin chain with nearest-neighbors interactions

“…This fact strongly suggests that the chain (5) should be regarded as the first instance of a quasi-exactly solvable spin chain, thus extending the usual notion of quasi-exact solvability to finite-dimensional Hamiltonians. It is interesting to note, in this respect, that the three lowest frequencies of the small oscillations near the equilibrium of the classical potential (31) are also integers [48].…”

Partially Solvable Spin Chains and QES Spin Models

“…It is interesting to note that the three lowest frequencies of the small oscillations near the equilibrium of the classical potential U 2 are also integers [15].…”

Polynomials associated with equilibria of affine Toda–Sutherland systems

“…At the equilibrium, the classical multi-particle dynamical system (2) is reduced to a set of harmonic oscillators. The frequencies (not frequencies squared) of small oscillations at the equilibrium of the affine Toda-Sutherland model are given up to the coupling constant g by [15] 1…”

“…The resulting polynomials for various affine root systems Π 0 are (we follow the affine Lie algebra notation used in [15,17]):…”

“…One is naturally led to a similar investigation for partially solvable or quasi-exactly solvable [13] multi-particle dynamics. From a not-so-long list of known quasi-exactly solvable multi-particle dynamical systems [14], we pick up the so-called affine Toda-Sutherland systems [15] and determine polynomials describing the equilibrium positions. These polynomials, as well as all the polynomials mentioned above, are characterised as having integer coefficients only.…”