We investigate low-density, quantum-degenerate gases in the presence of a localised attractive potential in the centre of a one-dimensional harmonic trap.The attractive potential is modelled using a parameterised delta-function, allowing us to determine all single particle eigenfunctions analytically. From these we calculate the ground state many-body properties for a system of spin-polarised fermions and, using the Bose-Fermi mapping theorem, extend the results to strongly interacting bosonic systems. We discuss the single particle densities, the pair correlation functions, the reduced single particle density matrices and the momentum distributions as a function of particle number and strength of the attractive point potential. As an important experimental observable, we place special emphasis on spatial coherence properties of such samples. ; Comment: 15 pages, 9 figures