Analytic Bethe Ansatz Solutions for Highest States in the su(1|1) and su(2) Sectors

We construct the integral equations by taking the thermodynamic limit of both the all-loop gauge Bethe ansatz equation and the string Bethe ansatz equation for the highest states in the su(1|1) and su(2) sectors of the N=4 super Yang-Mills theory. Using the Fourier transformation we solve the integral equations iteratively to obtain the anomalous dimensions of the highest states in the weak coupling expansion. In the su(1|1) sector we analytically study the strong coupling limit of the anomalous dimension for the all-loop gauge Bethe ansatz equation by means of the Laplace transformation