Phase Transition and Eigenvalue Degeneracy of a One Dimensional Anharmonic Oscillator

For a one dimensional lattice model with long range exponential interaction the problem of deriving the thermodynamic properties of the system reduces to the problem of determining the spectrum of a differential equation for a one dimensional anharmonic oscillator with potential αx² + βγx⁴ where γ is the inverse of the range of the interaction and α is a function of temperature which is positive at high temperatures and negative at low temperatures. It is proved that at low temperatures, α < 0, and in the limit γ → 0, (corresponding to an infinitely long range interaction) the spectrum becomes asymptotically degenerate. The relation of degeneracy to the occurrence of a phase transition is discussed and the generalization to higher dimensions is mentioned briefly.