Eigenvalues of the Schrödinger equation via the Riccati-Padé method

Fernández, Francisco M.; Ma, Q.; Tipping, R. H.

doi:10.1103/physreva.40.6149

Cited by 73 publications

(127 citation statements)

References 7 publications

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“…In particular the practical determination of the two adjustable parameters R and α, inherent to the method [see eqs. (34,36)], has been shown to correspond precisely to those analytic properties. In absence of any information on them, the criterion of best convergence is appeared to be valid.…”

Section: Discussionmentioning

confidence: 99%

“…The Padé method (named the Ricatti-Padé method in [34] and later on the Hankel-Padé method in [35]) has been first introduced in [10] in conjonction with a logarithmic-derivative transform like (8) to calculate, notably, the even and odd fundamental energies of the AO with β = 1 and m = 2 and for various values of λ. Typically the accuracy obtained was about 8 significant figures.…”

Section: The Padé Methodsmentioning

confidence: 99%

“…The distance to the origin of the closest singularity provides the value R 0 ≃ 5.192695 and the plane cut on the negative real axis, starting from the point z = −5.192695, forms an angular sector such that R = R 0 and α = α 0 = 2. Choosing those values in the conformal mapping (34,36), produces a truncated seriesg M (w) in powers of w. As M is increased the sum ofg M (w) at w = 1 should approach the asymptotic behavior corresponding to (6,15,16). Thus, accounting for those constraints, the auxiliary condition looked for to estimate E 0 may be expressed as:g (2) M (w) w=1 = 0 (37) in whichg (2) (w) stands for the conformal mapping applied on the function g ′′ (z) which, according to (6,15,16), goes to zero as z → ∞.…”

Section: 2mentioning

confidence: 99%

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Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

Bervillier¹

2009

J. Phys. A: Math. Theor.

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Abstract. The simplicity and the efficiency of a quasi-analytical method for solving nonlinear ordinary differential equations (ODE), is illustrated on the study of anharmonic oscillators (AO) with a potential V (x) = βx 2 + x 2m (m > 0). The method [Nucl. Phys. B801, 296 (2008)], applies a priori to any ODE with two-point boundaries (one being located at infinity), the solution of which has singularities in the complex plane of the independent variable x. A conformal mapping of a suitably chosen angular sector of the complex plane of x upon the unit disc centered at the origin makes convergent the transformed Taylor series of the generic solution so that the boundary condition at infinity can be easily imposed. In principle, this constraint, when applied on the logarithmic-derivative of the wave function, determines the eigenvalues to an arbitrary level of accuracy. In practice, for β ≥ 0 or slightly negative, the accuracy of the results obtained is astonishingly large with regards to the modest computing power used. It is explained why the efficiency of the method decreases as β is more and more negative. Various aspects of the method and comparisons with some seemingly similar methods, based also on expressing the solution as a Taylor series, are shortly reviewed, presented and discussed.

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Section: Discussionmentioning

confidence: 99%

Section: The Padé Methodsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

Bervillier¹

2009

J. Phys. A: Math. Theor.

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“…For instance, in a typical Dirichlet-Laplacian 1 Finding lower bounds for the smallest eigenvalue of a typical Hamiltonian is far much difficult than finding upper bounds. For successful attempts, see for instance the moment method proposed in [HB85], the Riccati-Padé method proposed in [FMT89] and the lower bounds obtained for few-body systems in [BMB + 98].…”

Section: Introductionmentioning

confidence: 99%

Upper and lower bounds for an eigenvalue associated with a positive eigenvector

Mouchet

2006

Journal of Mathematical Physics

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When an eigenvector of a semi-bounded operator is positive, we show that a remarkably simple argument allows to obtain upper and lower bounds for its associated eigenvalue. This theorem is a substantial generalization of Barta-like inequalities and can be applied to non-necessarily purely quadratic Hamiltonians. An application for a magnetic Hamiltonian is given and the case of a discrete Schrödinger operator is also discussed. It is shown how this approach leads to some explicit bounds on the ground-state energy of a system made of an arbitrary number of attractive Coulombian particles.

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“…In particular, though variational methods naturally provide upper bounds on E 0 , obtaining lower estimates requires more sophisticated techniques (for instance the Temple-like methods [19,§ XIII.2]), some of them being very system-dependent (e.g. the moment method proposed in [10] * mouchet@phys.univ-tours.fr for rational-fraction potentials or the Riccati-Padé method proposed in [7] for one-dimensional Schrödinger equations).…”

Section: Introductionmentioning

confidence: 99%

Bounding the ground-state energy of a many-body system with the differential method

Mouchet¹

2006

Nuclear Physics A

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This paper promotes the differential method as a new fruitful strategy for estimating a ground-state energy of a many-body system. The case of an arbitrary number of attractive Coulombian particles is specifically studied and we make some favorable comparison of the differential method to the existing approaches that rely on variational principles. A bird's-eye view of the treatment of more general interactions is also given.

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Eigenvalues of the Schrödinger equation via the Riccati-Padé method

Cited by 73 publications

References 7 publications

Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

Upper and lower bounds for an eigenvalue associated with a positive eigenvector

Bounding the ground-state energy of a many-body system with the differential method

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