  The transfer-matrix of the U(1) lattice model is considered in the Fourier
basis and in the weak coupling limit. The issues of Gauss law constraint and
gauge invariant states are addressed in the Fourier basis. In particular, it is
shown that in the strong coupling limit the gauge invariant Fourier states are
effectively the finite size closed loop currents. In the weak coupling limit,
however, the link-currents along periodic or infinite spatial directions find
comparable roles as gauge invariant states. The subtleties related to the
extreme weak coupling of the transfer-matrix in the Fourier basis are
discussed. A careful analysis of the zero eigenvalues of the matrix in the
quadratic action leads to a safe extraction of the diverging group volume in
the limit $g\to 0$. By means of the very basic notions and tools of the lattice
model, the spectrum at the weak coupling limit for any dimension and size of
lattice is obtained analytically. The spectrum at the weak coupling limit
corresponds to the expected one by the continuum model in the large lattice
limit.
