Generalized anharmonic oscillator: A simple variational approach

An approximate, variational method for the study of the generalized anharmonic oscillator $\frac{p^2}{2m}+\frac{m{\omega }^2{x}^2}{2}+\lambda V\left(x\right)\mathrm{}$ is presented, $V\left(x\right)\mathrm{}$ representing a broad class of even functions of the coordinate which admits a series expansion. The idea of the method is to introduce into the unperturbed oscillator states the correlations due to the presence of the anharmonic term via a unitary operator $e^{i\hat{F}}$ which is determined by the variational principle. Applications to the special case $\lambda x^{2m}$ are made, and the results are compared with those obtained with different exact treatments.