  Representing a strongly interacting multi-particle wave function in a finite
product basis leads to errors. Simple rescaling of the contact interaction can
preserve the low-lying energy spectrum and long-wavelength structure of wave
functions in one-dimensional systems and thus correct for the basis set
truncation error. The analytic form of the rescaling is found for a
two-particle system where the rescaling is exact. Detailed comparison between
finite Hilbert space calculations and exact results for up to 5 particles show
that rescaling can significantly improve the accuracy of numerical calculations
in various external potentials. In addition to ground state energies, the
low-lying excitation spectrum, density profile and correlation functions are
studied. The results give a promising outlook for numerical simulations of
trapped ultracold atoms.
