Accurate calculation of the eigenvalues of the potential

Fernández, Francisco M.; Mesón, Alejandro M.; Castro, Eduardo A.

doi:10.1016/0021-9991(83)90167-5

Cited by 5 publications

(4 citation statements)

References 15 publications

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“…The above estimates of E 0 and E 1 are not as accurate as those obtained using the Padé method with M ≃ 40 only [33]:…”

Section: Double Wellmentioning

confidence: 56%

“…In general it is sufficient to choose one of the three values ω = 1, 0, −1 according to whether the function to be determined goes to ±∞, a constant or 0 when z → ∞. Eventually, considering two successive values of ω gives upper and lower bounds on the eigenvalues [32,10,33]. The advantage of the Padé method is that it is well adapted to reproduce, from its Taylor expansion, the analytic structure of a meromorphic function especially if it has poles.…”

Section: The Padé Methodsmentioning

confidence: 99%

“…This method has then been utilized several times (see e.g. [33] for a list of references). In [33] the first two eigenvalues of both the quartic AO and the double well down to β = −15 are estimated with 20-18 significant figures.…”

Section: The Padé Methodsmentioning

confidence: 99%

“…[33] for a list of references). In [33] the first two eigenvalues of both the quartic AO and the double well down to β = −15 are estimated with 20-18 significant figures. In addition, Amore and Fernández [36] have shown that the Padé method may also be applied to solve the two-point boundary value problem associated to several nonlinear ODEs.…”

Section: The Padé Methodsmentioning

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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“…The above estimates of E 0 and E 1 are not as accurate as those obtained using the Padé method with M ≃ 40 only [33]:…”

Section: Double Wellmentioning

confidence: 56%

Section: The Padé Methodsmentioning

confidence: 99%

Section: The Padé Methodsmentioning

confidence: 99%

Section: The Padé Methodsmentioning

confidence: 99%

See 2 more Smart Citations

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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Accurate computation of the energy spectrum for potentials with multiminima

Taşeli

1993

Int J of Quantum Chemistry

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The eigenvalues of the Schrodinger equation with a polynomial potential are calculated accurately by means of the Rayleigh-Ritz variational method and a basis set of functions satisfying Dirichlet boundary conditions. The method is applied to the well potentials having one, two, and three minima. It is shown, in the entire range of coupling constants, that the basis set of trigonometric functions has the capability of yielding the energy spectra of unbounded problems without any loss of convergence providing that the boundary value a remains greater than a critical value a C r .Only the computation of the nearly degenerate states of multiwell oscillators requires dealing with a relatively large truncation order. 0 1993

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The influence of the boundedness of polynomial potentials on the spectrum of the Schrödinger equation

Taşeli

1992

Journal of Computational Physics

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Accurate calculation of the eigenvalues of the potential

Cited by 5 publications

References 15 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

Accurate computation of the energy spectrum for potentials with multiminima

The influence of the boundedness of polynomial potentials on the spectrum of the Schrödinger equation

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