Notes on the Pade Approximation for an Anharmonic Oscillator

Pade approximation is studied taking as an example the ground-state energy E-0(1, beta) of the anharmonic oscillator with x(2) + beta x(4) potential. The convergence and the other properties of the Pade approximant have so far been deduced from its relation to the Stieltjes function. However, E-0(1, beta) cannot be related to a Stieltjes function, but to the so-called once-subtracted Stieltjes function, so that we modify the theory of the Pade approximation, first showing that the same formula as the one for the Stieltjes function can be used also for the once-subtracted Stieltjes function. The properties of the modified Pade approximants are deduced and confirmed by numerical examples, establishing their error bounds at the same time. The Pade approximation is found to be superior in precision even for beta = 0.1 in lower orders than the perturbation theory of any orders. The Pade method is suggested to be applicable to a wider class of problems.