  Electrons in a multielectron bubble in helium form a spherical,
two-dimensional system coupled to the ripplons at the bubble surface. The
electron-ripplon coupling, known to lead to polaronic effects, is shown to give
rise also to Cooper pairing. A Bardeen-Cooper-Schrieffer (BCS) Hamiltonian
arises from the analysis of the electron-ripplon interaction in the bubble, and
values of the coupling strength are obtained for different bubble
configurations. The BCS Hamiltonian on the sphere is analysed using the
Richardson method. We find that although the typical ripplon energies are
smaller than the splitting between electronic levels, a redistribution of the
electron density over the electronic levels is energetically favourable as
pairing correlations can be enhanced. The density of states of the system with
pairing correlations is derived. No gap is present, but the density of states
reveals a strong step-like increase at the pair-breaking energy. This feature
of the density of states should enable the unambiguous detection of the
proposed state with pairing correlations in the bubble, through either
capacitance spectroscopy or tunneling experiments, and allow to map out the
phase diagram of the electronic system in the bubble.
