Summation of perturbation series of eigenvalues and eigenfunctions of anharmonic oscillators

A perturbation approach to compute the bound states of the Schrödinger equation HPsi=EPsi with H0+lambdaV and Psi(x=+/- infinity)=0 is studied. The approach involves solving the corresponding Dirichlet problem H(R)Psi(R)=E(R)Psi(R) on a finite interval [-R,R] by the Rayleigh-Schrödinger perturbation theory (RSPT). The method is based on the fact that E(R),Psi(R) converge to E,Psi as R--> infinity. The model problems to study the summability properties of the RSPT series E(R)= sum(infinity)(k=0)E((k))(R)lambda(k) are the anharmonic oscillators H=p(2)+x(2)+lambda(x)(2M), with M=2,3,4 for which the RSPT produces strongly divergent series E= sum(infinity)(k=0)E((k))lambda(k). The summation of the latter series with large lambda for the octic case is considered as an extremely challenging summation problem, in part, since it was rigorously proven that the Padé approximants cannot converge and the two-point Padé approximants, which combine information of the renormalized weak coupling and strong coupling expansions, give relatively good results. The calculations of this work show that the ordinary Padé approximants from the sole un-normalized E(R) series for the octic oscillator give accurate results with small or large lambda. The coefficients E((k))(R) are calculated with the eigenvalue series of an operator H(Rn), whose resolvent converges to that of H(R) as n--> infinity. The Padé approximants of the RSPT eigenfunction series Psi(R)= sum(infinity)(k=0)psi((k))(R)lambda(k) also provide accurate results for the octic oscillator.