  We study electron pairing in a one-dimensional (1D) fermion gas at zero
temperature under zero- and finite-range, attractive, two-body interactions.
The binding energy of Cooper pairs (CPs) with zero total or center-of-mass
momentum (CMM) increases with attraction strength and decreases with
interaction range for fixed strength. The excitation energy of 1D CPs with
nonzero CMM display novel, unique properties. It satisfies a dispersion
relation with \textit{two} branches: a\ phonon-like \textit{linear }excitation
for small CP CMM; this is followed by roton-like \textit{quadratic} excitation
minimum for CMM greater than twice the Fermi wavenumber, but only above a
minimum threshold attraction strength. The expected quadratic-in-CMM dispersion
\textit{in vacuo }when the Fermi wavenumber is set to zero is recovered for
\textit{any% } coupling. This paper completes a three-part exploration
initiated in 2D and continued in 3D.

Strongly Correlated Electrons