Nonpolynomial anharmonic oscillator

It is shown that for a nonpolynomial anharmonic correction of the form $\beta \frac{x^4}{4(1+a{x}^2)}$ to the simple harmonic oscillator with $H =p^2+\frac{1}{4}{x}^2$, the perturbation series in $\beta$ is convergent over a finite domain $\left|\beta \right|<a$. The energy eigenvalues have a two-sheeted structure with the cut extending from $\beta =-a$ to $\beta =-\infty$, and $\beta =-a$ is the accumulation point of singularities on the second sheet along real $\beta >-a$. An estimate for the perturbation series, using dispersion relation, is presented.