Energy-reflection symmetry of Lie-algebraic problems: Where the quasiclassical and weak-coupling expansions meet

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+l eigenfunctions in the algebraic sector are paired and inside each pair are related to each other by a simple analytic continuation x→ix, except the zero mode appearing if j is integer. At j→∞ 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. Both series coincide identically