Convergent Rayleigh–Schrödinger perturbation expansions for low-lying eigenstates and eigenenergies of anharmonic oscillators in intermediate basis states

Abstract

With suitably chosen unperturbed Hamiltonians, we show numerical evidence of convergence of Rayleigh–Schrödinger perturbation expansions for low-lying eigenstates and the corresponding eigenenergies of the quartic, sextic, and octic anharmonic oscillators, when the anharmonic terms are not very strong. In obtaining the perturbation expansions, unperturbed Hamiltonians are taken as the diagonal parts of the Hamiltonian matrices of the anharmonic oscillators in intermediate basis states and perturbations are taken as the off-diagonal parts. Intermediate basis states are calculated by part diagonalization of the total Hamiltonians in small subspaces of the underlying Hilbert space. In some strong-coupling regimes of the quartic and sextic anharmonic oscillators, the very simple approach of this Letter gives much more accurate results than previously used techniques.