View source for MIC-Kepler problem

''in classical mechanics''

The [[Hamiltonian system|Hamiltonian system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110120/m1101201.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110120/m1101202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110120/m1101203.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110120/m1101204.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110120/m1101205.png" /></td> </tr></table>

which can be considered as a one-parameter deformation family of the standard Kepler problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110120/m1101206.png" /> with the remarkable property that it retains its high dynamical symmetries. Physically, the deformation parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110120/m1101207.png" /> is interpreted as the magnetic charge of the particle at rest and measures the pitch of the cone on which trajectories lie. Its genuine mathematical interpretation is as a [[Cohomology|cohomology]] class of the symplectic structure. The global symmetry group of the problem is either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110120/m1101208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110120/m1101209.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110120/m11012010.png" />, depending on whether the energy is negative, zero or positive. The motion satisfies Kepler's three laws (cf. [[Kepler equation|Kepler equation]]).

The Hilbert spaces associated with the quantized problem carry almost all unitary irreducible representations of the respective covering groups, the only exception being the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110120/m11012011.png" />, for which only the [[Principal series|principal series]] representations arise. All this allows one to derive the spectrum and multiplicities of the bound states, as well differential cross sections of the scattering process and quantizations of the magnetic charge.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. McIntosh,   A. Cisneros,   "Degeneracy in the presence of magnetic monopole"  ''J. Math. Phys.'' , '''11'''  (1970)  pp. 896–916</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Mladenov,   V. Tsanov,   "Geometric quantization of the MIC-Kepler problem"  ''J. Phys. A: Math. Gen.'' , '''20'''  (1987)  pp. 5865–5871</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Mladenov,   "Scattering of charged particles off dyons"  ''J. Physics A Math. and Gen.'' , '''21'''  (1988)  pp. L1–L4</TD></TR></table>