  A K-matrix for waveguide confined spin-polarized fermionic atoms recently
computed by Granger and Blume is identified, in the low-energy domain, with a
contact condition for one-dimensional (1D) spinless fermions. Difficulties in
consistently formulating the contact conditions in terms of interaction
potentials are discussed and a rigorous alternative variational reformulation
is constructed. A duality between 1D fermions and bosons with zero-range
interactions suggested by Cheon and Shigehara is shown to hold for the
effective 1D dynamics of a spin-polarized Fermi gas with 3D p-wave interactions
and that of a Bose gas with 3D s-wave interactions in a tight waveguide. This
generalizes the mapping from impenetrable bosons (TG gas) to free fermions and
is used to derive the equation of state of an ultracold spin-polarized
fermionic vapor in a tight waveguide. Near a 1D confinement-induced resonance
one has a "fermionic TG gas" which maps to an ideal Bose gas.
