  The Bardeen-Cooper-Schrieffer (BCS)-Bose-Einstein condensation (BEC)
crossover in the two-dimensional attractive Hubbard model under the magnetic
field is discussed at the half-filling at T=0 K on the basis of the formalism
of Eagles and Leggett. It is shown that the so-called
Fulde-Ferrel-Larkin-Ovchinikov-like state with a non-zero center-of-mass wave
vector ${\bf q}\not=0$ is not stabilized in the weak-coupling (BCS) region,
while such a state with ${\bf q}\not=0$ is stabilized against that with ${\bf
q}=0$ even in a wide strong-coupling (BEC) region where di-fermion molecules
are formed. The physical implication of this surprising result is discussed.
