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 + 1 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. The both series coincide identically. † On leave of absence.