New exact solutions for polynomial oscillators in large dimensions

TL;DR

This paper introduces new exact solutions for polynomial oscillators in large dimensions, revealing explicit, closed-form solutions for certain polynomial potentials, which are fully factorizable and extend previous numerical approaches.

Contribution

It provides the first explicit, closed-form solutions for polynomial oscillators in large dimensions satisfying partial exact solvability conditions.

Findings

Explicit solutions exist for all N and q ≤ 5 in large dimensions.

Effective secular polynomials are fully factorizable over integers.

Contrasts with finite-dimensional models requiring numerical solutions.

Abstract

A new type of exact solvability is reported. We study the general central polynomial potentials (with 2q anharmonic terms) which satisfy the Magyari's partial exact solvability conditions (this means that they possess a harmonic-oscillator-like wave function proportional to a polynomial of any integer degree N). Working in the space of a very large dimension D for simplicity, we reveal that in contrast to the usual version of the model in finite dimensions (requiring a purely numerical treatment of the Magyari's constraints), our large D problem acquires an explicit, closed form solution at all N and up to q = 5 at least. This means that our effective secular polynomials (generated via the standard technique of Groebner bases) happen to be all fully factorizable in an utterly mysterious manner (mostly, over integers).