Collective modes of a spherical shell condensate

Dilute ultracold bosonic atoms in three-dimensional optical lattices with a harmonic confining potential are expected to exhibit inhomogeneous phases -- in particular phases containing superfluid regions confined to a spherical shell. It may also be possible to confine superfluids in this geometry using a special trap designed for this purpose. We explore the low-energy collective modes of a superfluid confined to a thin spherical shell analytically. In particular, in the limit of strong interactions we find two breathing modes with frequencies distinct from the spherical condensate breathing mode.