Abstract
A complete algebraic treatment of quantum problems with arbitrary central potential is given in terms of brokenO(4) invariance. This is done within the quasiclassical accuracy. More precisely, a generalised Runge-Lenz (RL) vector is built as a non-conserving member of theO(4) Poisson brackets algebra. Its algebraic partner is the orbital momentum. The requirements imposed on the RL vector are as follows: (i) its length is conserved, (ii) it rotates following the procession of the orbit, (iii) it vanishes for the circular motion. These requirements suffice for unique determination of the RL vector. This enables us to express the Hamiltonian of an arbitrary central problem as a function of theO(4) Casimir invariant and the angular momentum squared. The dependence upon the latter (generally very complicated) describes the way in which the symmetry is broken for a given potential. Replacing theO(4) Casimir operator and the angular momentum by their known eigenvalues results in the Bohr-Sommerfeld quantisation rules.O(4) multiplets of energy levels free of angular momentum degeneracy are described, examples are considered, and the inverse problem is discussed.
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