The one-boson Lee model is used as a simple example for demonstrating how the methods of dispersion theory can be used to calculate state vectors, wavefunction renormalization constants, and self-energies without having to solve eigenvalue equations. This method of solution has been applied successfully to the original Lee model; in this paper we give some analogous calculations at a more elementary level. The same point of view was taken in a previous paper where the Lehmann-Symanzik-Zimmermann formalism was used to solve the first nontrivial sector of the one-boson case.