Classical and quantum-mechanical systems of Toda lattice type. I

Abstract

The structure of the commutant of Laplace operators in the enveloping and "Poisson algebra" of certain generalized "αx + b" groups leads (in this article) to a determination of classical and quantum mechanical first integrals to generalized periodic and non-periodic Toda lattices. Certain new Hamiltonian systems of Toda lattice type are also shown to fit in this framework. Finite dimensional Lax forms for the (periodic) Toda lattices are given generalizing results of Flaschke.

On the complete integrability of the periodic quantum Toda lattice

“…The results are nicely summarized in [7,Theorem 3.3]. The importance of Theorem 1.1 resides in that it automatically implies the complete integrability of the periodic quantum Toda lattice associated to the extended Dynkin diagram ∆ ∪ {−β}, as explained in [6]. More recently, in the case when β is the highest root, the aforementioned integrability was established by Etingof in [3].…”

“…The theorem was proved in [6,7] in the cases when Π is of type A r , B r , C r , D r , or E 6 (note that when Π is of type B r or C r , there are two situations to be considered: when β is a long root and when β is a short root). The results are nicely summarized in [7,Theorem 3.3].…”

“…, u r of Sym(a) W ; by convention take u 1 := r i,j=1 α i , α j H i H j . The goal of this note is to prove the following result, which was conjectured by Goodman and Wallach in [6]. (We say that Γ ∈ U (b) has degree n if Γ ∈ U n (b) \ U n−1 (b), where U 0 (b) ⊂ U 1 (b) ⊂ .…”

“…We first recall the notion of "ax + b" Lie algebra, as defined by Goodman and Wallach in [6]. (In fact, the definition in that paper is more general: we only present here a special case which is relevant for our goal, which is to establish the complete integrability of the periodic quantum Toda lattice.)…”

Applications of loop groups and standard modules to Jacobians and theta functions of isospectral curves

“…Classical and quantum mechanical systems of Toda lattice type were studied in detail by R. Goodman and N. Wallach in a series of papers [6], [7], [8]. In the case of the generalized periodic Toda lattice the solution is calculated in terms of representative functions of standard modules of a Banach Lie group G w .…”