To characterize the complete structure of the Fucik spectrum of the p-Laplacian on higher
dimensional domains is a long-standing problem. In this paper, we study the p-Laplacian
with integrable potentials on an interval under the Dirichlet or the Neumann boundary
conditions. Based on the strong continuity and continuous differentiability of solutions
in potentials, we will give a comprehensive characterization of the corresponding Fucik
spectra: each of them is composed of two trivial lines and a double-sequence of differentiable,
strictly decreasing, hyperbolic-like curves; all asymptotic lines of these spectral
curves are precisely described by using eigenvalues of the p-Laplacian with potentials;
and moreover, all these spectral curves have strong continuity in potentials, i.e. as potentials
vary in the weak topology, these spectral curves are continuously dependent on
potentials in a certain sense.