A New Approximation Method for the Schrödinger Equation

Ighezou, F.-Z.; Lombard, R.J.

doi:10.1006/aphy.1999.5980

Cited by 17 publications

(25 citation statements)

References 16 publications

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“…A simple form of C (') is known to be exact for the Coulomb and the harmonic oscillator potentials, and to be a good first order approximation in many other cases [8,9]. The series (2) generalizes inequalities obtained by Bertlmann and Martin [10].…”

Section: Introductionmentioning

confidence: 92%

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Determination of the central density on the basis of its moments

et al. 2005

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Section: Introductionmentioning

confidence: 92%

“…It is valid in the case of a particle in a central potential, or two particles interacting via a scalar interaction. Details concerning its derivation and its application can be found in previous papers [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Determination of the central density on the basis of its moments

et al. 2005

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“…The purpose of this paper is to point out an interesting feature of the Schrödinger equation in the large-N limit. It concerns a series of recurrent relations first derived in N = 3 [32,33]. We shall show that they have the same formal expression in any dimensional space N 2.…”

mentioning

confidence: 87%

“…A priori, this factor is dependent on the potential, and nothing is gained. However, as stated in the introduction, it has been shown, in N = 3, that a simple empirical expression is very efficient for a wide class of potentials [32,33]. It has been constructed in a way to be exact for the Coulomb potential and the harmonic oscillator.…”

Section: The Recurrent Relationsmentioning

confidence: 99%

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Connection between the spectrum and the moments of the ground-state density inN-dimensional space

Al-Jaber¹,

Lombard

2005

J. Phys. A: Math. Gen.

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We show that a series of recurrent inequalities derived in N = 3 have the same formal expressions in any dimension N 2. They are derived from the multipole sum rules, and provide us with upper bounds for the moments of the ground-state density depending only on the lowest multipole excitation energy. These bounds are transformed into approximate recurrent relations by means of an empirical correction factor. The 1/r potential and the harmonic oscillator play a key role in establishing this factor, which is exact for these two potentials by construction. For a large class of potentials, we show that this factor tends to 1 as N → ∞. In such cases, at the large-N limit, the lowest state for each multipole excitation exhausts the sum rule. It thus acquires the characteristics of the one-phonon excitation typical of the harmonic oscillator.

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