  Using the Bethe ansatz analysis as was reformulated by Edwards, we calculate
the spectral properties of a particle interacting with a bath of fermions in
one dimension for the case of equal particle-fermion masses. These are directly
related to singularities apparent in optical experiments in one dimensional
systems. The orthogonality catastrophe for the case of an infinite particle
mass survives in the limit of equal masses. We find that the exponent $\beta$
of the quasiparticle weight, $Z\simeq N^{-\beta}$ is different for the two
cases, and proportional to their respective phaseshifts at the Fermi surface;
we present a simple physical argument for this difference. We also show that
these exponents describe the low energy behavior of the spectral function, for
repulsive as well as attractive interaction.
