  We use first-order perturbation theory near the fermionic limit of the
delta-function Bose gas in one dimension (i.e., a system of weakly interacting
fermions) to study three situations of physical interest. The calculation is
done using a pseudopotential which takes the form of a two-body
delta''-function interaction. The three cases considered are the behavior of
the system with a hard wall, with a point where the strength of the
pseudopotential changes discontinuously, and with a region of finite length
where the pseudopotential strength is non-zero (this is sometimes used as a
model for a quantum wire). In all cases, we obtain exact expressions for the
density to first order in the pseudopotential strength. The asymptotic
behaviors of the densities are in agreement with the results obtained from
bosonization for a Tomonaga-Luttinger liquid, namely, an interaction dependent
power-law decay of the density far from the hard wall, a reflection from the
point of discontinuity, and transmission resonances for the interacting region
of finite length. Our results provide a non-trivial verification of the
Tomonaga-Luttinger liquid description of the delta-function Bose gas near the
fermionic limit.
