New Type of Exact Solvability and of a Hidden Nonlinear Dynamical Symmetry in Anharmonic Oscillators

Schrödinger bound-state problem in D dimensions is considered for a set of central polynomial potentials containing 2q arbitrary coupling constants. Its polynomial (harmonicoscillator-like, quasi-exact, terminating) bound-state solutions of degree N are sought at an (q + 1)-plet of exceptional couplings/energies, the values of which comply with (the same number of) termination conditions. We revealed certain hidden regularities in these coupled polynomial equations and in their roots. A particularly impressive simplification of their pattern occurred at the very large spatial dimensions D 1 where all the “multi-spectra” of exceptional couplings/energies proved equidistant. In this way, one generalizes one of the key features of the elementary harmonic oscillators to (presumably, all) non-vanishing integers q > 0.