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The quantum Kepler problem is the study of a class of integrable
two body quantum systems,
the simplest of which is the hydrogen atom. It is shown that the
system defined on RD
has an so(2, D+1) dynamical symmetry, and that defined on the
superspace RD|2n has
a dynamical symmetry described by the orthosymplectic Lie superalgebra
osp(2, D+1|2n). In each case,
the negative energy eigenspaces span an infinite dimensional irreducible highest weight module for
the dynamical symmetry algebra. The module is constructed explicitly, and its weight spaces are
determined by studying the branching of the module with respect to so(D+1) or
osp(D+1|2n).
This in turn enables us to work out the spectrum of the quantum Hamiltonian, the corresponding
energy eigenspaces and their dimensions. Representation theoretical solutions of generalised
Kepler problems with magnetic monopole interactions will also be given.
Part of the material in this talk is based on joint work with Guowu Meng. |