Two-dimensional Dirac particles in a Pöschl-Teller waveguide

Hartmann, Richard; Portnoi, M. E.

doi:10.1038/s41598-017-11411-w

Cited by 45 publications

(47 citation statements)

References 94 publications

(122 reference statements)

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“…The Heun confluent equation [1][2][3] is a second order linear differential equation widely encountered in contemporary physics research ranging from hydrodynamics, polymer and chemical physics to atomic and particle physics, theory of black holes, general relativity and cosmology, etc. (see, e.g., [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and references therein). This equation has two regular singularities conventionally located at points 0 z  and 1 z  of complex z -plane, and an irregular singularity of rank 1 at z   .…”

Section: Introductionmentioning

confidence: 99%

Generalized confluent hypergeometric solutions of the Heun confluent equation

Ishkhanyan

2018

Applied Mathematics and Computation

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We show that the Heun confluent equation admits infinitely many solutions in terms of the confluent generalized hypergeometric functions. For each of these solutions a characteristic exponent of a regular singularity of the Heun confluent equation is a non-zero integer and the accessory parameter obeys a polynomial equation. Each of the solutions can be written as a linear combination with constant coefficients of a finite number of either the Kummer confluent hypergeometric functions or the Bessel functions.

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Section: Introductionmentioning

confidence: 99%

Generalized confluent hypergeometric solutions of the Heun confluent equation

Ishkhanyan

2018

Applied Mathematics and Computation

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“…The general Heun equation and its confluent forms appear in many fields of modern physics, such as general relativity, astrophysics, hydrodynamics, atomic and particle physics, etc. (see, e.g., [3,6,18,22,25,2,7]). A vast list of references to numerous physical applications, especially in general relativity, can be found in [9].…”

Section: Introductionmentioning

confidence: 99%

“…, n * − 1 are submitted to the recurrence (3): P n c n = Q n c n−1 + R n c n−2 , where P n , Q n , R n are defined by (4) and the initial conditions are c −1 = 0, c 0 = 1. From (7) and (8), we find that the coefficients s n for n = n * + 1, n * + 2, . .…”

Section: Introductionmentioning

confidence: 99%

On evaluation of the confluent Heun functions

Motygin

2018

2018 Days on Diffraction (DD)

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In this paper we consider the confluent Heun equation, which is a linear differential equation of second order with three singular points -two of them are regular and the third one is irregular of rank 1. The purpose of the work is to propose a procedure for numerical evaluation of the equation's solutions (confluent Heun functions). A scheme based on power series, asymptotic expansions and analytic continuation is described. Results of numerical tests are given. arXiv:1804.01007v1 [math.NA] 2 Apr 2018 2 Statement and basic notationsWe use the following form of the confluent Heun equation:This second order linear differential equation has regular singularities at z = 0 and 1, and an irregular singularity of rank 1 at z = ∞ (see e.g. [20]). The parameter q ∈ C is usually referred to as an accessory or auxiliary parameter and γ, δ, ε, α (also belonging to C) are exponent-related parameters.It is important to note that in this paper the parameters ε and α are assumed to be independent. Below, we will use notation c H (q, α, γ, δ, ε; z) or c H (z) for brevity. There are 6 local solutions of equation (1). The Frobenius method can be used to derive local power-series solutions to (1) near z = 0 and z = 1 (two per a singular point), while two solutions at z = ∞ can be obtained in the form of asymptotic series. In § 3 we will present the local solutions near the point z = 0. One of them is analytic in a vicinity of zero and if γ is not a nonpositive integer, we normalize this solution to unity at z = 0 and call it the local confluent Heun function. It is denoted by c Hl (q, α, γ, δ, ε; z). For the second Frobenius local solution, we will use the notation c Hs(q, α, γ, δ, ε; z). When γ is a nonpositive integer, one solution of (1) is analytic in a vicinity of z = 0 but it is equal to zero at z = 0, whereas the second solution can be normalized to unity at zero but generally it is not analytic. Following [16], the normalized solution will be denoted by c Hl (z) and another one by c Hs(z).

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“…Coloumb potential [19,20], Pöschl-Teller potential [20][21][22], Hulthén potential [23,24], Morse potential [25,26]. Among them as an example, Alhaidari et al [9] investigated the solutions in three dimensions with vector and scalar potentials, which are non-central, i.e.…”

Section: Introductionmentioning

confidence: 99%

Scattering of Klein-Gordon particles in the background of mixed scalar-vector generalized symmetric Woods-Saxon potential

2018

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Recently, it has been shown that the generalized symmetric Woods-Saxon potential energy, in which surface interaction terms are taken into account, describes the physical processes better than the standard form. Therefore in this study, we investigate the scattering of Klein-Gordon particles in the presence of both generalized symmetric Woods-Saxon vector and scalar potential. In one spatial dimension we obtain the solutions in terms of hypergeometric functions for spin symmetric or pseudo-spin symmetric cases. Finally, we plot transmission and reflection probabilities for incident particles with negative and positive energy for some critical arbitrary parameters and discuss the correlations for both cases.

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Two-dimensional Dirac particles in a Pöschl-Teller waveguide

Cited by 45 publications

References 94 publications

Generalized confluent hypergeometric solutions of the Heun confluent equation

Generalized confluent hypergeometric solutions of the Heun confluent equation

On evaluation of the confluent Heun functions

Scattering of Klein-Gordon particles in the background of mixed scalar-vector generalized symmetric Woods-Saxon potential

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