Analytical expressions for the energies of anharmonic oscillators

We propose a systematic construction of algebraic approximants for the boun d-state energies of anharmonic oscillators. The approximants are based on t he Rayleigh-Schrodinger perturbation series and take into account the analy tical behavior of the energies at large values of the perturbation paramete r. A simple expression obtained from a low-order perturbation series compar es favorably with alternative approximants. Present approximants converge i n the large-coupling limit and are suitable for the calculation of the ener gy of highly excited states. Moreover, we obtain some branch points of the eigenvalues of the anharmonic oscillator as functions of the coupling const ant.