Anharmonic Oscillator

Abstract

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