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

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.

Latex, 16 pages, 3 figures. Minor styllistic changes made, typos corrected, a remark on the energy-reflection symmetry in the quantum-algebraic Hamiltonians emerging in finite-difference problems added. Final version, to be published in Physical Review A

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