Quarkonium bound-state problem in momentum space revisited

Deloff, A.

doi:10.1016/j.aop.2006.10.004

Cited by 10 publications

(18 citation statements)

References 28 publications

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“…Chu et al [ 3,9 ] used the Legendre polynomials and Gauss‐Lobatto points in discretization and numerical integrations, and their so‐called GPS method have been extensively used in investigating the atomic and molecular structure calculations and laser‐atom interactions. Deloff [ 6,16 ] used the Chebyshev polynomials of the first kind and the classical Gauss points in discretization, and their so‐called semispectral method has wide applications in nuclear and particle physics. These two different frameworks have their own advantages, and their success has been established in the literature.…”

Section: Theorymentioning

confidence: 99%

“…The two options shown in Equations ) and () are completely equivalent because they are all discrete (shifted) delta functions. In the Chebyshev‐Gauss semispectral method developed by Deloff, [ 6,16 ] however, the first option was adopted by replacing L l and γ l with the Chebyshev polynomials and their corresponding normalization factors, respectively. In this work, we follow the latter one, that is, Equation ), as used in the work of Chu et al [ 3,9 ]…”

Section: Theorymentioning

confidence: 99%

“…[ 3 ] The GPS method can also be easily implemented to solve the Schrödinger equation in the momentum space, keeping in mind that solving such a singular problem in the momentum space is sometimes a formidable task for the basis expansion method. [ 16 ]…”

Section: Introductionmentioning

confidence: 99%

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Revisiting the generalized pseudospectral method: Radial expectation values, fine structure, and hyperfine splitting of confined atom

Zhu

Jiao

et al. 2020

Int J of Quantum Chemistry

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This paper revisits the generalized pseudospectral (GPS) method on the calculation of various radial expectation values of atomic systems, especially on the spatially confined hydrogen atom and harmonic oscillator. As one of the collocation methods based on global functions, the powerfulness and robustness of the GPS method has been well established in solving the radial Schrödinger equation with high accuracy.However, in our recent work, it was found that the previous calculations based on the GPS method for the radial expectation values of confined systems show significant discrepancies with other theoretical methods. In this work we have tackled such a problem by tracing its source to the GPS method and found that the method itself may not be able to obtain the system wave function at the origin. Combined with an extrapolation method developed here, the GPS method can fully reproduce the radial quantities obtained by other theoretical methods, but with more flexibility, efficiency, and accuracy. We apply the GPS-extrapolation method to investigate the relatievistic fine structure and hyperfine splitting of confined hydrogen atom in s-wave states where the zero-point wave function dominates. Good agreement with previous predictions is obtained for confined hydrogen in low-lying states, and benchmark results are obtained for high-lying excited states. The perturbation treatment of the fine and hyperfine interactions is validated in the confining environment. K E Y W O R D Sconfined atom, fine structure, generalized pseudospectral method, hyperfine splitting, radial expectation value | INTRODUCTIONIn the past few years, the generalized pseudospectral (GPS) method, which is also known as the semispectral method or collocation method, has been successfully applied to investigate the structural properties of atomic and molecular systems, [1][2][3][4][5] the quantum scattering processes in nuclear and atomic physics, [6,7] and the laser-atom interactions. [8][9][10] Its usefulness has been continuously revealed in recent years in accurately and efficiently solving both the time-independent and time-dependent Schrödinger and Dirac equations. Being a type of discrete variable representation (DVR) method, the GPS method shows its special superiority over other generalizations of DVR, such as the finite difference and finite element methods. For example, the Numerov and Runge-Kutta methods, as in the latter case, are local approaches to the unknown function by a sequence of overlapping low-order polynomials in a small subset of user-defined grid points. [11,12] The GPS method and its variants in different forms are generally global approaches to the unknown function using global basis functions with a high degree, for example, the trigonometric

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Section: Theorymentioning

confidence: 99%

Section: Theorymentioning

confidence: 99%

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Revisiting the generalized pseudospectral method: Radial expectation values, fine structure, and hyperfine splitting of confined atom

Zhu

Jiao

et al. 2020

Int J of Quantum Chemistry

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“…However, Ref. [24] did not pursue it further because it was considered not suitable for the approach proposed there.…”

Section: B Subtraction Of the Logarithmic Singularitymentioning

confidence: 99%

Linear confinement in momentum space: Singularity-free bound-state equations

et al. 2014

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Relativistic equations of Bethe-Salpeter type for hadron structure are most conveniently formulated in momentum space. The presence of confining interactions causes complications because the corresponding kernels are singular. This occurs not only in the relativistic case but also in the nonrelativistic Schrödinger equation where this problem can be studied more easily. For the linear confining interaction the singularity reduces to one of Cauchy principal value form. Although this singularity is integrable, it still makes accurate numerical solutions difficult. We show that this principal value singularity can be eliminated by means of a subtraction method. The resulting equation is much easier to solve and yields accurate and stable solutions. To test the method's numerical efficiency, we performed a three-parameter least-squares fit of a simple linear-plus-Coulomb potential to the bottomonium spectrum.

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“…During the last few decades, analytic and numerical studies [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] have been performed. In this paper, we employ the Landé subtraction method [28][29][30][31][32][33][34][35][36][37] to remove the momentum-space singularities [38] in the screened Cornell potential.…”

Section: Introductionmentioning

confidence: 99%

Extended Simpson’s rule for the screened Cornell potential in momentum space

Chen

2012

Phys. Rev. D

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We employ the Nyström method with the extended Simpson's rule to solve numerically the Schrödinger equation in momentum space with the screened Cornell potential. Two peculiar phenomena, furcation and deviation of the first point, occur in the obtained eigenfunctions. By applying the Landé subtraction method to remove or relieve the singularities in the integrands, the furcation phenomenon disappears, the deviation of the first point is weakened, and the obtained eigenvalues have high accuracy.

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Quarkonium bound-state problem in momentum space revisited

Cited by 10 publications

References 28 publications

Revisiting the generalized pseudospectral method: Radial expectation values, fine structure, and hyperfine splitting of confined atom

Revisiting the generalized pseudospectral method: Radial expectation values, fine structure, and hyperfine splitting of confined atom

Linear confinement in momentum space: Singularity-free bound-state equations

Extended Simpson’s rule for the screened Cornell potential in momentum space

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