Classical and quantum mechanical systems of Toda-lattice type

Relativistic Toda systems

“…In particular, the explicit formula (5.46) for the nonrelativistic case can be gleaned from Kostant's monumental work [9] by specializing to the root system AN-1 , cf. also [7,10,11]. We nevertheless present complete proofs for the nonrelativistic case, too.…”

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].…”

“…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].…”

Toda Flows and Isospectral Manifolds

“…There is such a unique decomposition (Iwasawa decomposition, see [11]). Then x(t) -Ad(Q(t)~x)-x0 is the solution to (4) such that x(o) -xo (for a proof see [10]). There is Qo e U such that Ad(ß0"1) -p = xn .…”

“…There is a unique element wm e W such that wm-A+ -A~ (see, e.g., [11]). In particular, the following Bruhat decomposition occurs: (10) G/73= (J BWm-(wo).…”