Quasi-exact solvability of Dirac equation with Lorentz scalar potential

In the present review, we deal with the recently introduced method of spectral parameter power series (SPPS) and show how its application leads to an explicit form of the characteristic equation for different eigenvalue problems involving Sturm–Liouville equations with variable coefficients. We consider Sturm–Liouville problems on finite intervals; problems with periodic potentials involving the construction of Hill's discriminant and Floquet–Bloch solutions; quantum‐mechanical spectral and transmission problems as well as the eigenvalue problems for the Zakharov–Shabat system. In all these cases, we obtain a characteristic equation of the problem, which in fact reduces to finding zeros of an analytic function given by its Taylor series. We illustrate the application of the method with several numerical examples, which show that at present, the SPPS method is the easiest in the implementation, the most accurate, and efficient. We emphasize that the SPPS method is not a purely numerical technique. It gives an analytical representation both for the solution and for the characteristic equation of the problem. This representation can be approximated by different numerical techniques and leads to a powerful numerical method, but most important, it offers a different insight into the spectral and transmission problems. Copyright © 2014 John Wiley & Sons, Ltd.

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“…Then and take the form of a Dirac system with a scalar potential (see, e.g., )

(\partial MathClass-bin+ q (x)) u MathClass-rel= λv MathClass-punc,

(\partial MathClass-bin- q (x)) v MathClass-rel= λu MathClass-punc.

…”

Section: Zakharov–shabat Eigenvalue Problemmentioning

confidence: 99%

Eigenvalue problems, spectral parameter power series, and modern applications

Khmelnytskaya

Kravchenko

Rosu

2014

Math Methods in App Sciences

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“…We are interested in the particular solution of the Dirac equation with the Yukawa interaction potential (4). In this section, we anticipate the answer and then obtain the parameters for highest nuclei, deuteron.…”

Section: Exact Analytical Solution Of the Dirac Equation For Yukawa Pmentioning

confidence: 99%

“…In particular, the Dirac equation which describes the motion of a spin-1/2 particle has been used in solving many problems of nuclear and high-energy physics. Recently, there has been an increase in searching for analytic solution of the Dirac equation [1][2][3][4][5][6][7][8][9][10][11]. Recently, tensor couplings have been used widely in the studies of nuclear properties [12][13][14][15][16][17][18][19][20][21][22], and they were introduced into the Dirac equation by substitution ⃗ → ⃗ − ⋅̂( ) [16,23], where is one of the particles, mass and refers to harmonic oscillator.…”

Section: Introductionmentioning

confidence: 99%

Exact Solution of the Dirac Equation for the Yukawa Potential with Scalar and Vector Potentials and Tensor Interaction

Azizi

Salehi

Rajabi

2013

ISRN High Energy Physics

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We present exact solutions of the Dirac equation with Yukawa potential in the presence of a Coulomb-like tensor potential. For this goal we expand the Yukawa form of the nuclear potential in its mesonic clouds by using Taylor extension to the power of seventh and bring out its simple form. In order to obtain the energy eigenvalue and the corresponding wave functions in closed forms for this potential (with great powers and inverse exponent), we use ansatz method. We also regard the effects of spin-spin, spin-isospin, and isospin-isospin interactions on the relativistic energy spectra of nucleon. By using the obtained results, we have calculated the deuteron mass. The results of our model show that the deuteron spectrum is very close to the ones obtained in experiments.

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“…The fundamental idea behind the quasi-exact solvability is the existence of a hidden dynamical symmetry. QES systems can be studied by two main approaches: the analytical approach based on the Bethe ansatz [14][15][16][17][18][19] and the Lie algebraic approach [10][11][12][13]. These techniques are of great importance because only a few number of problems in quantum mechanics can be solved exactly.…”

Section: Introductionmentioning

confidence: 99%

Exact Solutions of a Class of Double-Well Potentials: Algebraic Bethe Ansatz

Baradaran

Panahi

2017

Advances in High Energy Physics

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this article was funded by SCOAP 3 .Applying the Bethe ansatz method, we investigate the Schrödinger equation for the three quasi-exactly solvable double-well potentials, namely, the generalized Manning potential, the Razavy bistable potential, and the hyperbolic Shifman potential. General exact expressions for the energies and the associated wave functions are obtained in terms of the roots of a set of algebraic equations. Also, we solve the same problems using the Lie algebraic approach of quasi-exact solvability through the (2) algebraization and show that the results are the same. The numerical evaluation of the energy spectrum is reported to display explicitly the energy levels splitting.

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Quasi-exact solvability of Dirac equation with Lorentz scalar potential

Cited by 30 publications

References 35 publications

Eigenvalue problems, spectral parameter power series, and modern applications

Eigenvalue problems, spectral parameter power series, and modern applications

Exact Solution of the Dirac Equation for the Yukawa Potential with Scalar and Vector Potentials and Tensor Interaction

Exact Solutions of a Class of Double-Well Potentials: Algebraic Bethe Ansatz

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