Variational perturbation approach to compute the bound states of Schrödinger operators
Abstract
In previous works it was proved that the eigensolutions of the stationary Schrödinger equation H _{Ω}Ψ_{Ω} = E _{Ω}Ψ_{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} ^{k}β^{{k}}, Ψ_{Ω} = ∑^{∞} _{{k=0}}ψ^{{k}} _{Ω}β^{{k}} converge to those of the series E = ∑^{∞} _{{k=0}} E ^{k}β^{{k}}, Ψ = ∑^{∞} _{{k=0}}ψ^{{k}}β^{{k}} as Ω → R ^{N} when V is the regular perturbation of H^{0} and E _{Ω} ^{0} is an eigenvalue of H _{Ω} ^{0} that tends to a nondegenerate eigenvalue E ^{0} of H^{0}, 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} ^{k}β^{{k}}, = ∑^{∞} _{{k=0}}β^{{k}} from an equation H _{Ωm} = E _{Ωm} in a mdimensional 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 H^{0}. 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 RayleighSchrödinger 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.
 Publication:

Nuovo Cimento B Serie
 Pub Date:
 January 2004
 DOI:
 10.1393/ncb/i2003100601
 Bibcode:
 2004NCimB.119...41N