Phase-integral calculation of the energy levels of a quantal anharmonic oscillator

The phase-integral method developed by N. Fr\"oman and P. O. Fr\"oman is used for solving the quantal eigenvalue problem of an anharmonic oscillator with quartic anharmonicity. The generalized Bohr-Sommerfeld quantization condition up to the seventh-order phase-integral approximation is expressed explicitly in terms of complete elliptic integrals. Solving this quantization condition numerically, and comparing the results with recent very accurate numerical results obtained by Banerjee, we present curves exhibiting in a general way the accuracies of various orders of the phase-integral approximations. These curves clearly illustrate the utility of higher-order phase-integral approximations for the treatment of anharmonic oscillators.