Miloslav ZNOJIL
Nulear Physics Institute,

Vladimir P. GERDT, Denis A. YANOVICH
Laboratory of Information Technologies,
Joint Institute for Nuclear Research,
141980 Dubna, RUSSIA

Symbolic manipulations and exact solvability of anharmonic oscillators in large dimensions

Schrödinger bound-state problem in D dimensions is considered. For a set of central polynomial potentials depending on 2q arbitrary coupling constants, exceptional harmonic-oscillator-like solutions are known to be determined by the nonlinear set of N+q coupled polynomial equations for N+q ``selfconsistency" parameters. This problem becomes extremely complicated beyond N + q 5 where we tried to solve it via a sophisticated construction of the Janet bases in a degree-reverse-lexicographical ordering, followed by their conversion into the pure lexicographical bases. To our great surprise we revealed that a tremendous simplification of the solutions occurs in the domain of the large spatial dimensions D 1. Elementary solutions seem then available at many choices of q and N. This means that the use of the Janet bases helped us to discover a brand new class of the exactly solvable models in quantum mechanics. Their incomplete classification will be outlined, and also a few related new open questions concerning their ``hidden nonlinear symmetries" will be formulated.