  In the setting of the fractional quantum Hall effect we study the effects of
strong, repulsive two-body interaction potentials of short range. We prove that
Haldane's pseudo-potential operators, including their pre-factors, emerge as
mathematically rigorous limits of such interactions when the range of the
potential tends to zero while its strength tends to infinity. In a common
approach the interaction potential is expanded in angular momentum eigenstates
in the lowest Landau level, which amounts to taking the pre-factors to be the
moments of the potential. Such a procedure is not appropriate for very strong
interactions, however, in particular not in the case of hard spheres. We derive
the formulas valid in the short-range case, which involve the scattering
lengths of the interaction potential in different angular momentum channels
rather than its moments. Our results hold for bosons and fermions alike and
generalize previous results in \cite{LS}, which apply to bosons in the lowest
angular momentum channel. Our main theorem asserts the convergence in a
norm-resolvent sense of the Hamiltonian on the whole Hilbert space, after
appropriate energy scalings, to Hamiltonians with contact interactions in the
lowest Landau level.
