## Fachbereich Physik

- Double Scaling Limits, Airy Functions and Multicritical Behaviour in O(N) Vektor Sigma Models (1995)
- O(N) vector sigma models possessing catastrophes in their action are studied. Coupling the limit N - > infinity with an appropriate scaling behaviour of the coupling constants, the partition function develops a singular factor. This is a generalized Airy function in the case of spacetime dimension zero and the partition function of a scalar field theory for positive spacetime dimension.

- Double Scaling Limits and Catastrophes of the zerodimensional O(N) Vector Sigma Model: The A-Series (1994)
- We evaluate the partition functions in the neighbourhood of catastrophes by saddle point integration and express them in terms of generalized Airy functions.

- The critical O(N) sigma-model at dimension 2<d<4: Hardy-Ramanujan distribution of quasi-primary fields and a collective fusion approach (1994)
- The distribution of quasiprimary fields of fixed classes characterized by their O(N) representations Y and the number p of vector fields from which they are composed at N=infty in dependence on their normal dimension delta is shown to obey a Hardy-Ramanujan law at leading order in a 1/N-expansion. We develop a method of collective fusion of the fundamental fields which yields arbitrary qps and resolves any degeneracy.

- Construction of exactly solvable quantum models of Calogero and Sutherland type with translation invariant four-particle interactions (1998)
- 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.

- Exactly solvable dynamical systems in the neighborhood of the Calogero model (1999)
- 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 \(N\).

- Critical O(N) -vector nonlinear sigma-models: a resume of their field structure (1993)
- Abstract: 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. (To appear in the Proceedings of the XXII Conference on Differential Geometry Methods in Theoretical Physics, Ixtapa, Mexico, September 20-24, 1993)