We consider an exactly solvable Hamiltonian for bosons in one-dimension interacting through zero-range attractive forces, and construct a complete basis of its A-particle eigenstates. The structure of the single-particle spectral function in the removal domain is investigated, by taking the overlap of the A-particle ground state with the various excited states of the (A-1) system. In particular we study the contribution to the spectral function of the different break-up channels in the A-1 continuum, and compare the results to general statements available in the literature. It is shown that the asymptotic behavior in coordinate space does not agree with conventional assumptions. The relation to recent (e,e'p) experiments at large values of missing energy and momentum is pointed out.