Energy Reflection Symmetry of Lie-Algebraic Problems: Where the Quasiclassical and Weak Coupling Expansions Meet

TL;DR

This paper explores a class of one-dimensional Lie-algebraic problems where the spectrum exhibits a symmetry, showing that quasiclassical and weak coupling expansions for energy levels coincide exactly.

Contribution

It introduces a new class of Lie-algebraic problems with symmetric spectra and demonstrates the exact agreement between quasiclassical and weak coupling expansions.

Findings

The spectrum has a dynamical symmetry E -> -E.

Eigenfunctions are paired and related by analytic continuation.

Quasiclassical and weak coupling expansions for energies coincide exactly.

Abstract

We construct a class of one-dimensional Lie-algebraic problems based on sl(2) where the spectrum in the algebraic sector has a dynamical symmetry E -> - E. All 2j+1 eigenfunctions in the algebraic sector are paired, and inside each pair are related to each other by simple analytic continuation x -> ix, except the zero mode appearing if j is integer. At j-> infinity the energy of the highest level in the algebraic sector can be calculated by virtue of the quasiclassical expansion, while the energy of the ground state can be calculated as a weak coupling expansion. The both series coincide identically.