Duality and Self-Duality (Energy Reflection Symmetry) of Quasi-Exactly Solvable Periodic Potentials

TL;DR

This paper explores a duality transformation in quasi-exactly solvable periodic potentials, revealing a symmetry that links spectral properties and connects weak coupling with semiclassical regimes.

Contribution

It introduces a duality transformation for QES periodic potentials, establishing a connection between their spectra and energy-reflection symmetry.

Findings

Duality maps spectra of dual QES potentials.

Self-dual point exhibits energy-reflection symmetry.

Transformation links perturbative and nonperturbative sectors.

Abstract

A class of spectral problems with a hidden Lie-algebraic structure is considered. We define a duality transformation which maps the spectrum of one quasi-exactly solvable (QES) periodic potential to that of another QES periodic potential. The self-dual point of this transformation corresponds to the energy-reflection symmetry found previously for certain QES systems. The duality transformation interchanges bands at the bottom (top) of the spectrum of one potential with gaps at the top (bottom) of the spectrum of the other, dual, potential. Thus, the duality transformation provides an exact mapping between the weak coupling (perturbative) and semiclassical (nonperturbative) sectors.