  We propose an analytic framework to study the nonperturbative solutions of
Witten's open string field theory. The method is based on the Moyal star
formulation where the kinetic term can be split into two parts. The first one
describes the spectrum of two identical half strings which are independent from
each other. The second one, which we call midpoint correction, shifts the half
string spectrum to that of the standard open string. We show that the nonlinear
equation of motion of string field theory is exactly solvable at zeroth order
in the midpoint correction. An infinite number of solutions are classified in
terms of projection operators. Among them, there exists only one stable
solution which is identical to the standard butterfly state. We include the
effect of the midpoint correction around each exact zeroth order solution as a
perturbation expansion which can be formally summed to the complete exact
solution.
