The double exponential sinc collocation method for singular Sturm-Liouville problems

Gaudreau, Philippe; Slevinsky, Richard M.; Safouhi, Hassan

doi:10.1063/1.4947059

Cited by 16 publications

(16 citation statements)

References 23 publications

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“…We will primarily follow the procedure outlined in Ref. [13] and [20]. As in the HO basis, we decompose u α (r) into a linear combination of sinc basis functions:…”

Section: The Eigenvalue Problem In the Sinc Basismentioning

confidence: 99%

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Comparing Sinc and Harmonic Oscillator Basis for Bound States of a Gaussian Interaction

Sharaf

McCarty²,

Basili³

et al. 2020

SSRN Journal

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We investigate the use of the sinc collocation and harmonic oscillator bases for solving a two-particle system bound by a Gaussian potential described by the radial Schrödinger equation. We analyze the properties of the bound state wave functions by investigating where the basis-state wave functions break down and relate the breakdowns to the infrared and ultraviolet scales for both bases. We propose a correction for the asymptotic infrared region, the long range tails of the wave functions. We compare the calculated bound state eigenvalues and mean square radii obtained within the two bases. From the trends in the numerical results, we identify the advantages and disadvantages of the two bases. We find that the sinc basis performs better in our implementation for accurately computing both the deeply-and weakly-bound states whereas the harmonic oscillator basis is more convenient since the basis-state wave functions are orthogonal and maintain the same mathematical structure in both position and momentum space. These mathematical properties of the harmonic oscillator basis are especially advantageous in problems where one employs both position and momentum space. The main disadvantage of the harmonic oscillator basis as illustrated in this work is the large basis space size required to obtain accurate results simultaneously for deeply-and weakly-bound states. The main disadvantage of the sinc basis could be the numerical challenges for its implementation in a many-body application.

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“…We will primarily follow the procedure outlined in Ref. [13] and [20]. As in the HO basis, we decompose u α (r) into a linear combination of sinc basis functions:…”

Section: The Eigenvalue Problem In the Sinc Basismentioning

confidence: 99%

“…as used in Ref. [13] and [20]. We will need the cases i = 0 (zeroth derivative) and i = 2 (second derivative).…”

Section: The Eigenvalue Problem In the Sinc Basismentioning

confidence: 99%

Comparing Sinc and Harmonic Oscillator Basis for Bound States of a Gaussian Interaction

Sharaf

McCarty²,

Basili³

et al. 2020

SSRN Journal

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“…In [53,Theorem 3.2], we present the convergence analysis of DESCM which we state here in the case of the transformed Schrödinger equation (21). The proof of the Theorem is given in [53].…”

Section: General Definitions Properties and Preliminariesmentioning

confidence: 99%

“…It is the purpose of this research work to present such a general method based on the double exponential Sinc-collocation method (DESCM)presented in [53]. The DESCM starts by approximating the wave function as a series of weighted Sinc functions.…”

Section: Introductionmentioning

confidence: 99%

Double exponential sinc-collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function

Gaudreau

Safouhi

2017

Journal of Mathematical Physics

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In the present contribution, we apply the double exponential Sinc-collocation method (DESCM) to the one-dimensional time independent Schrödinger equation for a class of rational potentials of the form V (x) = p(x)/q(x). This algorithm is based on the discretization of the Hamiltonian of the Schrödinger equation using Sinc expansions. This discretization results in a generalized eigenvalue problem where the eigenvalues correspond to approximations of the energy values of the corresponding Hamiltonian. A systematic numerical study is conducted, beginning with test potentials with known eigenvalues and moving to rational potentials of increasing degree.

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“…The Sinc function and Sinc collocation method have been used extensively since their introduction to solve a variety of numerical problems [10][11][12]. The applications include numerical integration, linear and non-linear ordinary differential equations as well as partial differential equations [13][14][15][16][17][18][19][20][21][22]. The single exponential Sinc collocation method (SESCM) has been shown to offer an exponential convergence rate and works well in the presence of singularities.…”

Section: Introductionmentioning

confidence: 99%

On the Computation of Eigenvalues of the Anharmonic Coulombic Potential

2017

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In this work, we propose a method combining the Sinc collocation method with the double exponential transformation for computing the eigenvalues of the anharmonic Coulombic potential. We introduce a scaling factor that improves the convergence speed and the stability of the method. Further, we apply this method to Coulombic potentials leading to a highly efficient and accurate computation of the eigenvalues.

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The double exponential sinc collocation method for singular Sturm-Liouville problems

Cited by 16 publications

References 23 publications

Comparing Sinc and Harmonic Oscillator Basis for Bound States of a Gaussian Interaction

Comparing Sinc and Harmonic Oscillator Basis for Bound States of a Gaussian Interaction

Double exponential sinc-collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function

On the Computation of Eigenvalues of the Anharmonic Coulombic Potential

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