Nulear Physics Institute,

250 68 Rez, CZECH REPUBLIC

E-mail: znojil@ujf.cas.cz

*Vladimir P. GERDT,
Denis A. YANOVICH*

Laboratory of Information Technologies,

Joint Institute for Nuclear Research,

141980 Dubna, RUSSIA

E-mail: gerdt@jinr.ru, yan@jinr.ru

**Symbolic manipulations and exact solvability of anharmonic
oscillators in large dimensions**

**Abstract:**

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 ³ 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.