A Solvable Toy Model for Tachyon Condensation in String Field Theory

The lump solution of \\phi^3 field theory provides a toy model for unstable D-branes of bosonic string theory. The field theory living on this lump is itself a cubic field theory involving a tachyon, two additional scalar fields, and a scalar field continuum. Its action can be written explicitly because the fluctuation spectrum of the lump turns out to be governed by a solvable Schroedinger equation; the \\ell=3 case of a series of reflectionless potentials. We study the multiscalar tachyon potential both exactly and in the level expansion, obtaining insight into issues of convergence, branches of the solution space, and the mechanism for removal of states after condensation. In particular we find an interpretation for the puzzling finite domain of definition of string field marginal parameters.