We consider the anharmonic oscillator defined by the differential equation and the boundary condition limit of Φ(x) as x→±∞=0. This model is interesting because the perturbation series for the ground-state energy diverges. To investigate the reason for this divergence, we analytically continue the energy levels of the Hamiltonian H into the complex λ plane. Using WKB techniques, we find that the energy levels as a function of λ, or more generally of λ^α, have an infinite number of branch points with a limit point at λ=0. Thus, the origin is not an isolated singularity. Level crossing occurs at each branch point. If we choose α=1 / 3, the resolvent (z-H)^(-1) has no branch cut. However, for all z it has an infinite sequence of poles which have a limit point at the origin. The anharmonic oscillator is of particular interest to field theoreticians because it is a model of λϕ^4 field theory in one-dimensional space-time. The unusual and unexpected properties exhibited by this model may give some indication of the analytic structure of a more realistic field theory.