Approximation procedure for an anharmonic oscillator with cubic and quartic terms

We use -after a shift transformation of the variable- the Burrows, Cohen and Feldmann approximation procedure to solve the problem of finding the energy eigenvalues for an anharmonic oscillator with cubic and quartic terms subjected to a linear external potential. Both low- and high-frequency limits are considered. A first application is given by deriving (in the high-frequency case) the partition function of a gas composed of such anharmonic oscillators. We also exploit the recently proved formal equivalence between a high-frequency anharmonic oscillator (in the approximation considered) and an infinitesimally deformed harmonic oscillator to introduce SU(2) and SU(1, 1) algebras for the anharmonic oscillator with cubic and quartic terms.