Abstract
We consider the anharmonic oscillator defined by the differential equation and the boundary condition . 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 into the complex plane. Using WKB techniques, we find that the energy levels as a function of , or more generally of , have an infinite number of branch points with a limit point at . Thus, the origin is not an isolated singularity. Level crossing occurs at each branch point. If we choose , the resolvent has no branch cut. However, for all 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 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.
- Received 4 February 1969
DOI:https://doi.org/10.1103/PhysRev.184.1231
©1969 American Physical Society