  When a gas of electrons is confined to two dimensions, application of a
strong magnetic field may lead to startling phenomena such as emergence of
electron pairing. According to a theory this manifests itself as appearance of
the fractional quantum Hall effect with a quantized conductivity at an unusual
half-integer nu=5/2 Landau level filling. Here we show that similar electron
pairing may occur in quantum dots where the gas of electrons is trapped by
external electric potentials into small quantum Hall droplets. However, we also
find theoretical and experimental evidence that, depending on the shape of the
external potential, the paired electron state can break down, which leads to a
fragmentation of charge and spin densities into incompressible domains. The
fragmentation of the quantum Hall states could be an issue in the proposed
experiments that aim to probe for non-abelian quasi-particle characteristics of
the nu=5/2 quantum Hall state.
