  We consider an electronic spin in a quantum dot, coupled to the surrounding
nuclear spins via inhomogeneous antiferromagnetic hyperfine interactions and
subject to a uniform field, which is described by Gaudin's central spin model.
We study spectral properties, the two-point correlation functions, and the
magnetization profile in the ground state and in low-lying excited states,
which characterizes the structure of the cloud of nuclear spins screening the
electron spin. A close connection to the pair occupation probability in the
BCS-model is established. Using the exact Bethe Ansatz solution of that model
and arguments of integrability, we can distinguish between contributions from
purely classical physics and from quantum fluctuations.
