The eigenvalue problem of one-dimensional Dirac operator

Karwowski, Jacek; Ishkhanyan, А. М.; Poszwa, A.

doi:10.1007/s00214-020-02685-x

Cited by 7 publications

(5 citation statements)

References 67 publications

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“…( ) ¢ W x . Indeed, a rather similar effective Schrödinger potential is also obtained in the case of pure vector and scalar potentials (see, e.g., [21,22]). For instance, in the case of a scalar interaction potential ( ) S x , one arrives at a Schrödinger-like equation (for an auxiliary function, see [23]) with the effective potential…”

Section: Potentialmentioning

confidence: 64%

“…In this case, it should be taken into account that we have two components of the wave function. Examining then equation (21), we notice that the wave function for the energies given by equation (24) diverges at the origin. Therefore, these energies should be discarded.…”

Section: Dirac Casementioning

confidence: 96%

“…Let us discuss the Schrödinger equation (8) for the wave function y 1R for the Stillinger potential (13), without referring to the y 2R and, therefore, the hints coming from equation (21). Based on general considerations [25], we must require that the wave function at the origin behaves like / y ~x , 1R 5 6 so we require = A 0.…”

Section: Schrödinger Casementioning

confidence: 99%

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Conditionally exactly solvable Dirac potential, including x ^1/3 pseudoscalar interaction

Ishkhanyan

Крайнов

2023

Phys. Scr.

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We investigate the properties of a conditionally exactly solvable Dirac potential, including an interaction term proportional to and a term with a fixed strength. Using a combination of analytical and numerical methods, we analyze the energy spectrum and wavefunctions for this potential. Our results show that the interplay between the two interaction terms leads to properties that are not present in the case of commonly discussed Dirac potentials. These findings may have useful implications for our understanding of quantum systems and contribute to our understanding of the structure of the Dirac equation.

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Section: Potentialmentioning

confidence: 64%

Section: Dirac Casementioning

confidence: 96%

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Conditionally exactly solvable Dirac potential, including x ^1/3 pseudoscalar interaction

Ishkhanyan

Крайнов

2023

Phys. Scr.

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“…This fact is not surprising since the Dirac equation D Q ψ = Eψ leads to two decoupled equations involving a quadratic pencil of the Schrödinger operator, one per each spinor component, cf. [3]. More precisely, the referred equations read…”

Section: A Square Electrostatic Potentialmentioning

confidence: 99%

Energy Power Series Analysis of the Bound States of the One-dimensional Dirac Equation

Barrera-Figueroa

2023

J. Phys.: Conf. Ser.

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In this work we consider the one-dimensional Dirac equation including an electrostatic potential with compact support, and focus on the regime of bound states. We obtain exact expressions for both the characteristic function and the eigenfunctions in L 2 (ℝ, ℂ2), given in the form of power series of the energy parameter. This approach is meant for arbitrary bounded potentials, so that a square potential is a special case of the theory here presented. We derive an efficient numerical method for the calculation of approximate eigen-energies of the bound states. Finally, we investigate the physical sense of the eigen-energies that are forbidden in the non-relativistic regime in terms of the Klein tunneling.

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“…During the last two decades, one-dimensional Dirac equation has been studied extensively. Application of this equation range from solid-state physics to nuclear and elementary particle physics [1][2][3][4][5][6]. Also, Dirac operators attracted significant consideration, in the spectral theory [7,8], non-linear Schrödinger equations [9], or as an effective model for graphene [10,11].…”

mentioning

confidence: 99%

Pseudospectral method for solving the fractional one-dimensional Dirac operator using Chebyshev cardinal functions

et al. 2023

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In this paper, a numerical method is introduced to ﬁnd the eigenvalues and eigenfunctions of the Caputo fractional Dirac operator. To this end, the problem reduces to a Volterra integral equation with a weakly singular kernel. Then, the pseudospectral method based on Chebyshev cardinal functions is used to solve the obtained Volterra integral equation. By introducing the operational matrix of the fractional integral operator for cardinal Chebyshev functions, the Volterra integral equation is reduced to an algebraic system. To obtain the approximation of the eigenvalues, it is suﬃcient to ﬁnd the roots of the characteristic function of the algebraic system. Then, the convergence of the method is proved. To demonstrate the ability and accuracy of the method some numerical examples are solved.

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The eigenvalue problem of one-dimensional Dirac operator

Abstract: The properties of the eigenvalue problem of the one-dimensional Dirac operator are discussed in terms of the mutual relations between vector, scalar and pseudo-scalar contributions to the potential. Relations to the exact solubility are analyzed.

Cited by 7 publications

References 67 publications

Conditionally exactly solvable Dirac potential, including x ^1/3 pseudoscalar interaction

Conditionally exactly solvable Dirac potential, including x ^1/3 pseudoscalar interaction

Energy Power Series Analysis of the Bound States of the One-dimensional Dirac Equation

Pseudospectral method for solving the fractional one-dimensional Dirac operator using Chebyshev cardinal functions

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The eigenvalue problem of one-dimensional Dirac operator

Abstract: The properties of the eigenvalue problem of the one-dimensional Dirac operator are discussed in terms of the mutual relations between vector, scalar and pseudo-scalar contributions to the potential. Relations to the exact solubility are analyzed.

Cited by 7 publications

References 67 publications

Conditionally exactly solvable Dirac potential, including x 1/3 pseudoscalar interaction

Conditionally exactly solvable Dirac potential, including x 1/3 pseudoscalar interaction

Energy Power Series Analysis of the Bound States of the One-dimensional Dirac Equation

Pseudospectral method for solving the fractional one-dimensional Dirac operator using Chebyshev cardinal functions

Contact Info

Product

Resources

About

Conditionally exactly solvable Dirac potential, including x ^1/3 pseudoscalar interaction

Conditionally exactly solvable Dirac potential, including x ^1/3 pseudoscalar interaction