The logarithmic corrections in the one-dimensional Hubbard model with attraction

The one-dimensional Hubbard model is considered. The ground state energy as the function of the density (chemical potential) in the vicinity of the half-filled band is calculated. For the model defined on the finite-size lattice with $N$ sites the decomposition of the elementary excitation energy is obtained with the accuracy up to $(N^2\ln N)^{-1}$. The explicit expression for the free energy and the spectrum of elementary excitations as functions of the external fields or the volume $N$ is necessary for the investigation the long distance asymptotics of the correlation functions.