Analytic Structure of Energy Levels in a Field-Theory Model

We consider a model of $\lambda {\varphi }^4$ field theory in which perturbation theory diverges and analytically continue the energy levels into the complex $\lambda$ plane. Using WKB technique, we determine that the energy levels have an infinite sequence of branch points (where level crossing occurs) with a limit point $a\lambda =0\mathrm{}$. Thus the origin is not an isolated singularity. The resolvent ${\mathrm{}(z-H)\mathrm{}}^{-1\mathrm{}}$ has an infinite sequence of poles with a limit point at $\lambda =0\mathrm{}$.