Anharmonic Oscillator

We consider the anharmonic oscillator defined by the differential equation $(-\frac{d^2}{d{x}^2}+\frac{1}{4}{x}^2+\frac{1}{4}\lambda x^4)\Phi \mathrm{}\left(x\right)=E\left(\lambda \right)\Phi \mathrm{}\left(x\right)\mathrm{}$ and the boundary condition $limit\; of\Phi \mathrm{}\left(x\right)\mathrm{}\text{as}x\to \pm \infty =0\mathrm{}$. 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 $\lambda$ plane. Using WKB techniques, we find that the energy levels as a function of $\lambda$, or more generally of ${\lambda }^{\alpha }$, have an infinite number of branch points with a limit point at $\lambda =0\mathrm{}$. Thus, the origin is not an isolated singularity. Level crossing occurs at each branch point. If we choose $\alpha =\frac{1}{3}$, the resolvent ${\mathrm{}(z-H)\mathrm{}}^{-1\mathrm{}}$ 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 $\lambda {\varphi }^4$ 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.