Double Scaling Limits and Catastrophes of the zerodimensional O(N) Vector Sigma Model: The A-Series

We evaluate the partition functions in the neighbourhood of catastrophes by saddle point integration and express them in terms of generalized Airy functions.Comment: 20 pages, LaTeX, two figures available from author on reques

An exactly solvable model of the Calogero type for the icosahedral group

We construct a quantum mechanical model of the Calogero type for the icosahedral group as the structural group. Exact solvability is proved and the spectrum is derived explicitly.Comment: 13 pages, no figures, latex 2epsilo

Critical O(N) - vector nonlinear sigma - models: A resume of their field structure

The classification of quasi - primary fields is outlined. It is proved that the only conserved quasi - primary currents are the energy - momentum tensor and the O(N) - Noether currents. Derivation of all quasi - primary fields and the resolution of degeneracy is sketched. Finally the limits d=2 and d=4 of the space dimension are discussed. Whereas the latter is trivial the former is only almost so.Comment: 16 pages, Latex. To appear in the Proceedings of the XXII Conference on Differential Methods in Theoretical Physics, Ixtapa, Mexico, September 20-24, 199

The construction of trigonometric invariants for Weyl groups and the derivation of corresponding exactly solvable Sutherland models

Trigonometric invariants are defined for each Weyl group orbit on the root lattice. They are real and periodic on the coroot lattice. Their polynomial algebra is spanned by a basis which is calculated by means of an algorithm. The invariants of the basis can be used as coordinates in any cell of the coroot space and lead to an exactly solvable model of Sutherland type. We apply this construction to the $F_4$ caseComment: 13 pages, no figures, latex 2epsilon, corrected versio

Exactly solvable dynamical systems in the neighborhood of the Calogero model

The Hamiltonian of the $N$-particle Calogero model can be expressed in terms of generators of a Lie algebra for a definite class of representations. Maintaining this Lie algebra, its representations, and the flatness of the Riemannian metric belonging to the second order differential operator, the set of all possible quadratic Lie algebra forms is investigated. For N=3 and N=4 such forms are constructed explicitly and shown to correspond to exactly solvable Sutherland models. The results can be carried over easily to all $N$.Comment: 23 pages, 2 figures, replaced and enlarged versio

Construction of exactly solvable quantum models of Calogero and Sutherland type with translation invariant four-particle interactions

We construct exactly solvable models for four particles moving on a real line or on a circle with translation invariant two- and four-particle interactions.Comment: 14 pages, no figures, 1 table, replaced and enlarged version, a note adde