New type of exact solvability and of a hidden nonlinear dynamical symmetry in anharmonic oscillators

TL;DR

This paper uncovers a new exact solvability and hidden nonlinear symmetry in anharmonic oscillators, revealing regularities in polynomial solutions and spectra, especially at large dimensions, extending harmonic oscillator properties to complex potentials.

Contribution

It introduces a novel approach to solving anharmonic oscillators by identifying hidden symmetries and regularities in polynomial solutions and spectra at large dimensions.

Findings

Polynomial solutions exist at exceptional couplings/energies

Spectra become equidistant at large spatial dimensions

Hidden nonlinear dynamical symmetry is revealed

Abstract

Schroedinger bound-state problem in D dimensions is considered for a set of central polynomial potentials (containing 2q coupling constants). Its polynomial (harmonic-oscillator-like, quasi-exact, terminating) bound-state solutions of degree N are sought at a (q+1)-plet of exceptional couplings/energies, the values of which comply with (the same number of) termination conditions. We revealed certain hidden regularity in these coupled polynomial equations and in their roots. A particularly impressive simplification of the pattern occurred at the very large spatial dimensions D 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.