  We consider a system of spinless fermions in a nearly one-dimensional
cylindrical trap. We introduce four-fermion interaction terms through which
particles in low-lying modes of the trap may interact with particles in
radially or angularly excited states. Treating these interactions as
perturbations about a purely one-dimensional Luttinger liquid, we calculate the
zero-temperature energy shifts of the bosonic quanta of the Luttinger model. We
perform this calculation is two ways. First we use the bosonized forms of the
fermion operators. Then we introduce a simplified, physically-motivated model;
in this model, the methods used immediately generalize, so that we may also
calculate the energy shifts of the Tomonaga-Luttinger model. By equating the
results from the two disparate models, we obtain the numerical value of the
Luttinger liquid bandwidth for this system.
