Based on the Bogoliubov non-ideal gas model, we discuss the energy spectrum
and phase transition of the superfluid Fermi gas of atoms with a weak
attractive interaction on the canonical noncommutative space. Because the
interaction of a BCS-type superfluid Fermi gas originates from a pair of
Fermionic quasi-particles with opposite momenta and spins, the Hamiltonian of
the Fermi gas on the noncommutative space can be described in terms of the
ordinary creation and annihilation operators related to the commutative space,
while the noncommutative effect appears only in the coefficients of the
interacting Hamiltonian. As a result, we can rigorously solve the energy
spectrum of the Fermi gas on the noncommutative space exactly following the way
adopted on the commutative space without the use of perturbation theory. In
particular, different from the previous results on the noncommutative
degenerate electron gas and superconductor where only the first order
corrections of the ground state energy level and energy gap were derived, we
obtain the nonperturbative energy spectrum for the noncommutative superfluid
Fermi gas, and find that each energy level contains a corrected factor of
cosine function of noncommutative parameters. In addition, our result shows
that the energy gap becomes narrow and the critical temperature of phase
transition from a superfluid state to an ordinary fluid state decreases when
compared with that in the commutative case
