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 s1( 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 ins ide 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-->infinity the energy of the highest level in the algebraic sector can be calculated by vi rtue of the quasiclassical expansion, while the energy of the ground state can be calculated as a weak-coupling expansion. Both series coincide identi cally. [S1050-2947(99)00703-9].