We extend a previous numerical study of a one-dimensional generalized Wick-Cutkosky model in which complex scalars interact via the exchange of real scalars. The numerical techniques are based on discretized light-cone quantization and the Lanczos diagonalization algorithm. Contrary to previous results, we find that the vacuum instability of a cubic theory can be detected numerically, given adequate resolution. For comparison with the unstable case, we also consider two stable cases: one where a Tamm-Dancoff truncation is applied and another where a quartic interaction of sufficient strength is added. In each case we compute the mass gap and the critical coupling where the gap disappears.