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The Kepler problem is a physics problem about two bodies
which attract each other by a force obeying the inverse square law.
Historically, the solution of this problem at the classical level (the
quantum level respectively) gave a
satisfactory explanation of the planetary motion ( the spectral lines of the
hydrogen atom respectively). It is well-known that the Kepler problem is a
super-integrable model; but it is much less known that there is a big family
of super-integrable models of the Kepler type.
In constructing a model in this
big family, we specify a compact reductive Lie group G (the gauge group)
together with an irreducible representation s (the magnetic charge) for G.
This model M(G, s) has a non-compact dynamical
symmetry group G' and its Hilbert space of bound states (denoted by s') is an
unitary highest weight representation of G'. Consequently, we obtain a
correspondence (G, s) <---> (G', s') and a simple geometric/physical
realization of s'.
Currently, the super-integrable
model with (G, G') being any of the following pairs:
(Spin(n), Spin(2, n+2)), (O(1), Sp^~(2n, R)), (U(1), U^~(n,n)), (Sp(1),
O^*(4n))
has been constructed. It turns out that the correspondence is the theta
correspondence when (G, G') is any of the last three pairs.
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