  We present an inverse scattering construction of generalised point
interactions (GPI) -- point-like objects with non-trivial scattering behaviour.
The construction is developed for single centre $S$-wave GPI models with
rational $S$-matrices, and starts from an integral transform suggested by the
scattering data. The theory of unitary dilations is then applied to construct a
unitary mapping between Pontryagin spaces which extend the usual position and
momentum Hilbert spaces. The GPI Hamiltonian is defined as a multiplication
operator on the momentum Pontryagin space and its free parameters are fixed by
a physical locality requirement. We determine the spectral properties and
domain of the Hamiltonian in general, and construct the resolvent and M{\o}ller
wave operators thus verifying that the Hamiltonian exhibits the required
scattering behaviour. The physical Hilbert space is identified. The
construction is illustrated by GPI models representing the effective range
approximation. For negative effective range we recover a known class of GPI
models, whilst the positive effective range models appear to be new. We discuss
the interpretation of these models, along with possible extensions to our
construction.
