Time-Compact Scheme for the One-Dimensional Dirac Equation

Cao, Jun-Jie; Li, Xiang-Gui; Jing-liang, Qiu; Zhang, Jingjing

doi:10.1155/2016/3670139

Cited by 4 publications

(5 citation statements)

References 19 publications

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“…A full momentum space approach is given in [13]. Nonetheless, most approaches involving the momentum space combine it with the real space via the split-operator formalism [14,15,16,17,18,19,20,21]. These numerical schemes approximate the time evolution operator by a sequence of operators that depend on either the momentum or the spatial variables, as both types of operators have an analytical expression in their respective spaces.…”

Section: Introductionmentioning

confidence: 99%

A conservative fourth-order real space method for the (2+1)D Dirac equation

Vanderstraeten

Ginste

2023

Journal of Computational and Applied Mathematics

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Section: Introductionmentioning

confidence: 99%

A conservative fourth-order real space method for the (2+1)D Dirac equation

Vanderstraeten

Ginste

2023

Journal of Computational and Applied Mathematics

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“…Although nonconstant potentials may imply challenging issues, spectral estimates can be obtained 4–7 . Similarly, the search for solutions may involve the use of sophisticated techniques 8–18 . In this work, we employ a method 19 that provides us with exact solutions in the form of power series involving the spectral parameter, which can be used in spectral problems.…”

Section: Introductionmentioning

confidence: 99%

“…[4][5][6][7] Similarly, the search for solutions may involve the use of sophisticated techniques. [8][9][10][11][12][13][14][15][16][17][18] In this work, we employ a method 19 that provides us with exact solutions in the form of power series involving the spectral parameter, which can be used in spectral problems.…”

Section: Introductionmentioning

confidence: 99%

On the calculation of the discrete spectra of one‐dimensional Dirac operators

Barrera-Figueroa

Rabinovich

Loredo‐Ramírez

2022

Math Methods in App Sciences

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In this work, we consider the one‐dimensional Dirac operator where is Pauli's matrix, is a ‐matrix representing a regular potential that includes the electrostatic and scalar interactions as well as the anomalous magnetic momentum, is a singular potential consisting of delta distributions ( ) are ‐matrices representing the strengths of Dirac deltas, and is a two‐spinor. We associate to the operator an unbounded in symmetric operator denoted by , where is the support of singular potential . The operator includes only the regular potential together with certain interaction conditions at each point . The paper presents a method for determining the discrete spectrum of the operator for arbitrary potential whose entries are given by ‐functions. The eigenvalues of the operator are the zeros of a dispersion equation , where the characteristic function is determined explicitly in terms of power series involving the spectral parameter . The construction of the characteristic function from a set of monodromy matrices and the interaction conditions is presented in the paper. Moreover, its power series representation leads to an efficient numerical method for calculating the eigenvalues of the Dirac operator from the zeros of certain approximate function which is obtained by truncating the series up to a finite number of terms. Several examples show the applicability and accuracy of the numerical method.

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“…Different parameter regimes could be considered for the Dirac equation (1.7) (or (1.17 of bound states and/or standing wave solutions, we refer to [26,27,32,40,41,52] and references therein. In this parameter regime, for the numerical part, many efficient and accurate numerical methods have been proposed and analyzed [3], such as the finite difference time domain (FDTD) methods [4,50], time-splitting Fourier pseudospectral (TSFP) method [9,42,20], exponential wave integrator Fourier pseudospectral (EWI-FP) method [9], the Gaussian beam method [62], etc.…”

Section: Introductionmentioning

confidence: 99%

A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation

Bao

Yin

2018

Res Math Sci

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We propose a new fourth-order compact time-splitting (S 4c ) Fourier pseudospectral method for the Dirac equation by splitting the Dirac equation into two parts together with using the double commutator between them to integrate the Dirac equation at each time interval. The method is explicit, fourth-order in time and spectral order in space. It is unconditional stable and conserves the total probability in the discretized level. It is called a compact time-splitting method since, at each time step, the number of sub-steps in S 4c is much less than those of the standard fourth-order splitting method and the fourth-order partitioned Runge-Kutta splitting method. Another advantage of S 4c is that it avoids to use negative time steps in integrating sub-problems at each time interval. Comparison between S 4c and many other existing time-splitting methods for the Dirac equation are carried out in terms of accuracy and efficiency as well as long time behavior. Numerical results demonstrate the advantage in terms of efficiency and accuracy of the proposed S 4c . Finally we report the spatial/temporal resolutions of S 4c for the Dirac equation in different parameter regimes including the nonrelativistic limit regime, the semiclassical limit regime, and the simultaneously nonrelativisic and massless limit regime.

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Time-Compact Scheme for the One-Dimensional Dirac Equation

Cited by 4 publications

References 19 publications

A conservative fourth-order real space method for the (2+1)D Dirac equation

A conservative fourth-order real space method for the (2+1)D Dirac equation

On the calculation of the discrete spectra of one‐dimensional Dirac operators

A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation

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