  We derive analytic expressions for the two-body matrix elements in finite
spherical quantum Hall systems in terms of a general scalar interaction
expressed as a sum over Legendre polynomials, and we derive the corresponding
pair pseudopotentials from these matrix elements. The relationship between the
effective spatial potential and the pseudopotential is one-to-one in this
framework, and we show how any complete model pseudopotential can be
analytically inverted to give a unique corresponding spatial potential. As an
example, we find the spatial potential that produces a harmonic pseudopotential
and verify that it fails to break the angular momentum degeneracy of the
many-body quantum Hall system. We also include additional examples to
demonstrate the use of the inversion technique.
