  The Bethe ansatz equations of the 1-D Hubbard model under open boundary
conditions are systematically derived by diagonalizing the inhomogeneous
transfer matrix of the XXX model with open boundaries. Through the finite-size
correction, we obtain the energy spectrum of the open chain and discuss the
effects of boundary magnetic fields applied only at the edges of the chain.
Several physical implications of the finite-size spectrum are discussed from
the viewpoint of the boundary conformal field theories (BCFT); the conformal
dimension of the spin excitation is quite sensitive to the boundary magnetic
fields, and this sensitivity can be explained in terms of the $\pi/2$-phase
shift of the BCFT. For several limiting cases such as for the half-filling
case, the conformal dimensions are explicitly calculated.
