Variational perturbation approach to compute the bound states of Schrödinger operators

In previous works it was proved that the eigensolutions of the stationary Schrödinger equation H _ΩΨ_Ω = E _ΩΨ_Ω in a finite box Ω ⊂ R ^N with impenetrable walls, converge to the bound states of the equation HΨ = EΨ in L ₂(R ^N) as Ω → R ^N. In this work we consider H = H ⁰ + βV and it is proved that i) the coefficients of the standard perturbation series E _Ω = ∑^∞ _{k=0} E _Ω ^kβ^{k}, Ψ_Ω = ∑^∞ _{k=0}ψ^{k} _Ωβ^{k} converge to those of the series E = ∑^∞ _{k=0} E ^kβ^{k}, Ψ = ∑^∞ _{k=0}ψ^{k}β^{k} as Ω → R ^N when V is the regular perturbation of H⁰ and E _Ω ⁰ is an eigenvalue of H _Ω ⁰ that tends to a nondegenerate eigenvalue E ⁰ of H⁰, and ii) the coefficients E _Ω ^k, ψ^{k} _Ω can be approximated by means of those of the series E _Ωm = ∑^∞ _{k=0} E _Ωm ^kβ^{k}, = ∑^∞ _{k=0}β^{k} from an equation H _Ωm = E _Ωm in a m-dimensional space when V is a bounded operator. In many cases of interest the series from H _Ω and H _Ωm are convergent for small |β|. This solves some problems posed by the asymptotic character of the E and Ψ series when V is a singular perturbation of H⁰. The summability properties of the series from H _Ω and H _Ωm are exhibited in a previous work by the Padé summation of the E _Ωm and series of the octic oscillator H _Ω = p ² + x ² + βx ^2M for small and large β values. The eigenstates of H _Ω and H _Ωm can be computed by means of the Rayleigh-Schrödinger perturbation theory even when H has a finite number of bound states. The results provide a theoretical framework to compute the eigenstates of spatially confined quantum systems.