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

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 2. 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. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'. (paper)