  We study a method for solving the homogeneous Bethe-Salpeter equation. By
introducing a `fictitious' eigenvalue $\lambda$ the homogeneous Bethe-Salpeter
equation is interpreted as a linear eigenvalue equation, where the bound state
mass is treated as an input parameter. Using the improved ladder approximation
with the constant fermion mass, we extensively study the spectrum of the
fictitious eigenvalue $\lambda$ for the vector bound states and find the
discrete spectrum for vanishing bound state mass. We also evaluate the bound
state masses by tuning appropriate eigenvalues $\lambda$ to be unity, and find
massless vector bound states for specific values of the constant fermion
masses.
