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{leftrightarrow}{minus}E. All 2j+1 eigenfunctions in the algebraic sector are paired and inside each pair are related to each other by a simple analytic continuation x{r_arrow}ix, except the zero mode appearing if {ital j} is integer. At j{r_arrow}{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. Both series coincide identically. {copyright} {ital 1999} {ital The American Physical Society}