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

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.  2002 Elsevier Science B.V. All rights reserved. PACS: 03.65.-w; 02.30.Lt; 02.70.-c In Rayleigh–Schrödinger (RS) perturbation theory, an eigenvalue of a Hamiltonian H is expressed in a perturbation expansion in power of some perturbation parameter λ. In most cases, perturbation expansions give divergent results [1]. Various summation methods, e.g., the Borel and Padé methods which are suitable to cases of not large λ, have been developed to extract physically meaningful results from divergent expansions [2,3]. In recent years, techniques such as δ expansion [4], variational perturbation theory [5], and E-mail address: [email protected] (W.-g. Wang). renormalized strong coupling expansion [6,7], have been developed for large λ. For the quartic, sextic, and octic anharmonic oscillators, which are well suited to illustrate the problem of divergent expansions, these techniques give results with impressive accuracy for ground-state energies. The techniques can also be generalized to calculate ground-state eigenfunctions [8], but, the accuracy of the results obtained is much lower than for ground-state energies. Recently, both the ordinary [9,10] and the renormalized strong coupling expansion methods have been made use of in the investigation of low-lying excited-states of anharmonic oscillators [11,12]. 0375-9601/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S0375-9601(02)0 12 29 -X 126 W.-g. Wang / Physics Letters A 303 (2002) 125–130 Various partition methods have been used in summation techniques in separating a Hamiltonian H into a solvable part H0 and a perturbation V , H =H0 +V . For example, V may be chosen by the principle of minimal sensitivity [13]. It is also possible for H0 to be chosen to have a tri-diagonal matrix form, which has been shown to be useful in the study of the sextic anharmonic oscillators with double-well shapes [14]. Usually, H0 is chosen to be analytically solvable. However, the analytical solvability of H0 is in fact unnecessary for the purpose of summation techniques, which is to calculate eigenenergies and eigenfunctions (if applicable) as accurate as possible. Indeed, in both the ordinary and the renormalized strong coupling expansion methods, the most singular part of the total Hamiltonian H is included in the unperturbed Hamiltonian H0, which makes it possible to construct convergent perturbation expansions, guaranteed by Kato boundedness [15,16]. In such divisions of the total Hamiltonian, the most difficult problem is the calculation of the expanding coefficients, since even numerical solution of the eigenenergies and eigenvectors of H0 is usually quite difficult. For the case of anharmonic oscillators, efficient techniques for the evaluation of the coefficients have been developed in recent years, which have confirmed the convergence of the expansions numerically [11,12,17]. The purpose of this Letter is to investigate the possibility of constructing convergent perturbation expansions, for which numerical solution of H0 can be carried out easily. For this purpose, the unperturbed Hamiltonian H0 is obtained by part diagonalization of the total Hamiltonian H in finite dimensional subspaces of the underlying Hilbert space. Suppose H has a partition (H (a) 0 ,V ), with H 0 being analytically solvable. A new partition (H0,V ) can be achieved by part diagonalization of H in small subspaces of the Hilbert space, each of which is spanned by a small number of the eigenstates of H 0 . At least in some cases, the geometric mean of |fkk′ | associated with H0 and V , where fkk′ = Vkk′/(E 0 − E0 k ) with E 0 k being eigenenergies of H0 and Vkk′ being couplings, can be smaller than that of |f (a) nn′ | associated with H 0 and V , and the sign of fkk′ can be more irregular than the sign of f (a) nn′ . Since RS perturbation expansions are mainly composed of products of factors like fkk′ , it is of interest to investigate whether the partition (H0,V ) could be better than (H (a) 0 ,V ), in giving useful perturbation expansions. In this Letter, for the quartic, sextic, and octic anharmonic oscillators, we will show numerically that, with suitably chosen H0 by part diagonalization of the total Hamiltonians, it is possible for the RS perturbation expansions to give convergent results for low-lying eigenenergies and eigenstates, when the anharmonic terms are not very strong. The method used here is easier to be carried out numerically than the methods such as the variational perturbation theory and the renormalized strong coupling expansions. Another advantage of the method is that it gives eigenstates with an accuracy similar to that of the eigenenergies. Let us first discuss the quartic anharmonic oscillator with a Hamiltonian (1) H = 1 2 p2 + 1 2 x2 +μx4, or written in terms of creation and annihilation operators, a+ = (x − ip)/√2 and a = (x + ip)/√2, H = a+a + 1 2 + 3 4 μ ( 2 ( a+a )2 + 2a+a + 1)