Quasi-exact solvability, resonances and trivial monodromy in ordinary differential equations

TL;DR

This paper explores quasi-exact solvability in higher-order differential equations, revealing links with Bender-Dunne polynomials and resonances, and extends known phenomena from second-order to more complex operators.

Contribution

It introduces a novel correspondence between sextic anharmonic oscillators and third-order ODEs, advancing understanding of quasi-exact solvability in higher-order differential operators.

Findings

Links with Bender-Dunne polynomials established

Resonances between solutions identified

Extended quasi-exact solvability to higher-order equations

Abstract

A correspondence between the sextic anharmonic oscillator and a pair of third-order ordinary differential equations is used to investigate the phenomenon of quasi-exact solvability for eigenvalue problems involving differential operators with order greater than two. In particular, links with Bender-Dunne polynomials and resonances between independent solutions are observed for certain second-order cases, and extended to the higher-order problems.