  A systematic method to obtain strong coupling expansions for scattering
quantities in Hamiltonian lattice field theories is presented. I develop the
conceptual ideas by means of the Hamiltonian field theory analogue of the Ising
model, in $d$ space and one time dimension. The main result is a convergent
series representation for the sacttering states and the transition matrix. To
be explicit the special cases of $d=1$ and $d=3$ spatial dimensions are
discussed in detail. I compute the next-to-leading order approximation for the
phase shifts. The application of the method to investigate low-energy
scattering phenomena in lattice gauge theory and QCD is proposed.
