Solution of Spin and Pseudo-Spin Symmetric Dirac Equation in (1+1) Space-Time Using Tridiagonal Representation Approach
Assi I. A.1, *, Alhaidari A. D.2, Bahlouli H.2, 3
Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John’s, NL A1B3X7, Canada
Saudi Center for Theoretical Physics, P.O. Box 32741, Jeddah 21438, Saudi Arabia
Physics Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

 

† Corresponding author. E-mail: iassi@mun.ca

Abstract
Abstract

The aim of this work is to find exact solutions of the Dirac equation in (1+1) space-time beyond the already known class. We consider exact spin (and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus (and minus) the vector potential. We also include pseudo-scalar potentials in the interaction. The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis, which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric. This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.

1 Introduction

The Dirac wave equation is used to describe the dynamics of spin one-half particles at high energies (but below the threshold of pair creation) in relativistic quantum mechanics. It is a relativistically covariant linear first order differential equation in space and time for a multi-component spinor wavefunction. This equation is consistent with both the principles of quantum mechanics and the theory of special relativity.[13] The physics and mathematics of the Dirac equation are very rich, illuminating and gave birth to the theoretical foundation for different physical phenomena that were not observed in the non-relativistic regime. Among others, we can cite the prediction of electron spin, the existence of antiparticles and tunneling through very high barriers, the so-called Klein tunneling.[46] In addition, Dirac equation appears at a lower energy scale in graphene (2-D array of carbon atoms), wherein the behavior of electrons is modeled by 2-D massless Dirac equation, the so-called Dirac-Weyl equation.[710] Recent relevant applications of the Dirac equation could be found in Refs. [1116] where the relativistic rotational and vibrational energy spectra are obtained for various physical systems. However, despite its fundamental importance in physics, exact solutions of the Dirac equation were obtained only for a very limited class of potentials.[1725]

In this paper, we study situations with spin or pseudo-spin symmetry, which are SU(2) symmetries of the Dirac equation that have different applications especially in nuclear physics.[2633] The spin symmetric case is generally defined for situations where Δ(r) = S(r) − V(r) = Cs, where Cs is a real constant while S(r) and V(r) are the scalar and vector components of the potential, respectively. Spin symmetry has been used to explain the suppression of spin-orbit splitting of meson states with heavy and light quarks. Pseudospin symmetry occurs when ∑(r) = S(r) + V(r) = Cp, where Cp is a real constant parameter. This latter symmetry was used to explain the near degeneracy of some single particle levels near the Fermi surface. Here, we will restrict our study to the exact symmetry where Cs = Cp = 0, that is when S(r) = ± V(r). Aside from their physical applications, these symmetries allow the decoupling of the upper and lower spinor components of the Dirac equation transforming it into a Schrödinger-like equation for each of the two components. This makes it mathematically easier to obtain analytic solutions of the original wave equation for certain potential configurations. In addition to the scalar and vector potentials, we also include a pseudo-scalar component to the potential configuration

Exact solutions of the Dirac equation are of great benefit both from the theoretical and applied point of view. Analytic solutions allow for a better understanding of physical phenomena and establish the necessary correspondence between relativistic effects and their non-relativistic analogues. In this spirit, we would like to revisit the one-dimensional Dirac equation and investigate all potentially solvable class of interactions using the tridiagonal representation approach (TRA).[3435] The hope is to be able to enlarge the conventional class of solvable potentials of the Dirac equation.

The organization of this work goes as follows. We give a review of the TRA in the next section. In Sec. 3, we present a mathematical formulation of the problem for the spin and pseudo-spin symmetric situations. Then, in Secs. 4 and 5 we present different examples of solvable potentials. Lastly, we conclude our work in Sec. 6.

2 Review of the TRA

The basic idea of the TRA is to write the spinor wavefunction as a bounded infinite series with respect to a suitably chosen square integrable basis functions. That is, |ψϵ(x)〉 = ∑m fm(ϵ)|ϕm(x)〉, where is a set of expansion coefficients that are functions of the energy ϵ and potential parameters whereas and is a complete set of properly chosen spinor basis functions. The stationary wave equation reads (Hϵ)|ψ〉 = J|ψ〉 = 0, where H is the Dirac Hamiltonian. We require that the matrix representation of the wave operator, Jn, m = 〈ϕn|(Hϵ)|ϕm〉, be tridiagonal and symmetric so that the action of the wave operator on the elements of the basis is allowed to take the general form (HE)|ϕn〉 ~ |ϕn〉 + |ϕn − 1〉 + |ϕn + 1〉. To achieve this requirement, we were obliged to use the kinetic balance equation that relates the upper and the lower spinor basis components transforming the wave equation into the following three-term recursion relation for :

Thus, the problem now is reduced to solving this three-term recursion relation, which is equivalent to solving the original problem since contain all physical information (both structural and dynamical) about the system. Of course, there are different mathematical techniques to solve this algebraic equation.[3637] For example, Eq. (1) could be written in a form that allows for direct comparison to well-known orthogonal polynomials. However, in other situations this recursion relation does not correspond to any of the known orthogonal polynomials hence giving rise to new classes of orthogonal polynomials. The remaining challenge will then be to extract physical information (e.g., energy spectrum and phase shift) from the properties of the associated orthogonal polynomials such as the weight function, generating function, spectrum formula, asymptotics, zeroes, etc.[3839]

The most general square-integrable basis that are used in the TRA take the following form[3435]

where y = y(x), Am is a normalization constant and Pm(y) is a polynomial of a degree m in y. Whereas, w(y) is a positive function that vanishes on the boundaries of the original configuration space with coordinate x and has a form, for convenience, similar to the weight function associated with the polynomial Pm(y). In our present work, we will be using two sets of bases:

The Laguerre basis, where are the Laguerre polynomials with y ⩾ 0, w(y) = yα e−βy, and {α, β, ν} are real parameters.

The Jacobi basis where are the Jacobi polynomials with y ∈ [−1, +1], w(y) = (1 − y)α(1 + y)β, and {α, β, μ, ν} are real and will be chosen so that we obtain a tridiagonal representation for the wave operator.

3 Formulation

In relativistic units, ħ = c = 1, the most general linear massive Dirac equation in (1+1) space-time dimension with time-independent potentials reads as follows:[13]

where {γμ}μ = 0, 1 are the two Dirac gamma matrices such that (γ0)2 = 1, (γ1)2 = −1, and γ0 γ1 = −γ1 γ0. S is the scalar potential, Aμ = (V, U) is the (time, space) component of the two-vector potential, W is the pseudo-scalar potential, and γ5 = iγ0γ1. Choosing the minimum dimensional representation for the gamma matrices defined by

gives

and makes the Dirac equation (3) take the following matrix form

where ϵ is the energy and we wrote the two-component spinor wavefunction as

The space component of the vector potential, U, could be eliminated by the local gauge transformation ψ(x) → e−iΛ(x)ψ(x) such that dΛ/dx = U. Therefore, from now on and for simplicity, we take U = 0. It should be noted that in (3+1) space-time with spherical symmetry, Eq. (4) with W(x) → W(x) + κ/x represents the radial Dirac equation with x being the radial coordinate and the spin-orbit quantum number κ = ± 1, ± 2, ± 3, … Now, the exact spin and pseudo-spin symmetric coupling correspond to S = V and S = −V, respectively. We discuss below the positive energy solution of the spin symmetric coupling in the time-independent Dirac equation. The negative energy pseudo-spin symmetric solution follows from the spin symmetric one by a straightforward map, which will be derived below.

Now, for spin symmetric coupling the Dirac equation (4) with U = 0 reads as follows

Giving the following relation between the two spinor components for the positive energy solution space

where ϵ ≠ −M. Substituting this expression of ψ−(x) in the first equation of Eq. (5) gives the following Schrödinger-like second order differential equation for the upper spinor component

The objective now is to find a discrete square integrable spinor basis in which the matrix representation of the wave equation (5) becomes tridiagonal and symmetric so that the corresponding three-term recursion relation could be solved exactly for the expansion coefficients of the wavefunction and for as large a class of potentials as possible. As noted in the introduction above, we write , where is a complete set of square integrable basis elements for the two wavefunction components and {fn(ϵ)} is an appropriate set of energy dependent functions. Now, in the TRA, we impose the requirement that the matrix representation of the Dirac wave operator Jnm = 〈ϕn|(Hϵ)| ϕm〉 is tridiagonal and symmetric (with ϕn being a spinor whose components are ) so that the wave equation (5) becomes a three-term recursion relation for the expansion coefficients {fn}:

In line with the kinetic balance approach for the two spinor components, we relate the two components of the spinor basis using Eq. (6) as . Using this and the fact that vanishes at the boundaries of configuration space, then we can perform integration by parts and rewrite the above equation as

Now, for the pseudo-spin symmetric coupling (S = −V), the Dirac equation (4) with U = 0 becomes

and Eq. (6) is replaced by

where ϵ ≠ +M. Substituting this in the second equation of Eq. (10) gives the following Schrödinger-like second order differential equation for the lower spin component

Comparing Eqs. (11) adn (12) with the corresponding spin symmetric case, we obtain the following map

Applying this map to Eq. (5), we obtain

Multiplying this equation by −1 then from left by , and noting that , we obtain

Giving an identical equation to the pseudo-spin symmetric Dirac equation (10). Thus, applying the map (13) on the positive energy spin symmetric solution gives the negative energy pseudo-spin symmetric solution.

In the following two sections, we obtain the exact positive energy solution of the spin symmetric Dirac equation (5) in the Laguerre and Jacobi bases by giving the expansion coefficients {fn} in terms of orthogonal polynomials in the energy variable. The asymptotics of these polynomials give the phase shift of the continuous energy scattering states and the spectrum of the discrete energy bound states.

4 Solution in the Laguerre Basis

Let y(x) be a transformation from the real configuration space with coordinate x to a new dimensionless coordinate y such that y ⩾ 0. A complete set of square integrable functions as basis for the wavefunction in the new y-space that also satisfy the desirable boundary conditions (vanish at the boundaries) could be chosen as follows

where and the real parameters are such that β > 0 and ν > −1. Substituting Eq. (16) into Eq. (6), as applied to the basis set, and using the differentiation chain rule d/dx = y′(d/dy), we obtain

where the prime stands for the derivative with respect to x. It is required that in function space must be nearest neighbor to . That is, the tridiagonal requirement on Eq. (17) means that should be expressed as a linear combination of terms in and . The differential property of the Laguerre polynomial, and its recursion relation, show that this could be achieved if we impose the constraint that be a linear function in y. That is, (y/y′)W(y) = ρy + σ where ρ and σ are dimensionless potential parameters. Thus, we can rewrite Eq. (17) as follows

In the following two subsections, we consider the two {possible} scenarios found in Appendix A that correspond to Eq. (A6a) and (A6b), respectively.

4.1 The q = 0 Scenario of Eq. (A6a)

In this scenario, the vector potential is V(y) = y2a − 1(A + By) and the pseudo-scalar potential is W(y) = λya − 1(ρy + σ), where a, A, and B are real parameters introduced in the Appendix A. Moreover, the basis parameters are ν2 = (2σ + 1 − a)2 and 2α = ν + 1 − a. There are two configurations in this scenario: one corresponds to a = 0 where y(x) = λx and the other corresponds to a = 1/2 where y(x) = (λx/2)2.

For the first configuration, we can write

where we have replaced x by the radial coordinate r and Z stands for the electric charge whereas κ is the spin-orbit coupling, which assumes the values ±1, ±2, ±3, … Therefore, the parameters in the Dirac wave operator matrix (A7a) are as follows:

These parameter assignments are physically motivated and will be supported by the results obtained below. Additionally, to make the vector potential pure Coulombic and vanish at infinity we can freely choose V0 = 0. Substituting these parameters in Eq. (A7a), then the Dirac equation H|ψ〉 = ϵ|ψ〉 results in the following symmetric three-term recursion relation for the expansion coefficients of the wave function

where . Dividing by and writing fn(ϵ) = f0(ϵ)Pn(ϵ), makes this a recursion relation for Pn(ϵ) with P0(ϵ) = 1. We compare it to that of the orthonormal version of the two-parameter Meixner-Pollaczek polynomial that reads

where the Meixner-Pollaczek polynomial is defined as

with z ∈ [−∞, +∞], μ > 0 and 0 < θ < π. Thus, we conclude that where

and

requiring that . The phase shift is obtained using the asymptotics (n → ∞) formula of the Meixner-Pollaczek polynomial. It reads δμ(z) = arg Γ(μ + iz) + μ(θπ/2) giving

The continuous energy wavefunction becomes . On the other hand, the spectrum formula of the Meixner-Pollaczek polynomial that reads gives the following relativistic energy spectrum

which is a quadratic equation that could easily be solved for ϵn. With W0 = 0, it is identical to the energy spectrum of the spin-symmetric Dirac-Coulomb problem (i.e., with equal scalar and vector Coulomb potentials). The mth bound state energy wavefunction, will be written in terms of the Meixner polynomial , which is the discrete version of the Meixner-Pollaczek polynomial, as . The orthonormal version of this polynomial is defined as

where 0 < γ < 1 and μ > 0. It satisfies the following recursion relation

where γ−1 = cosh(θ).

For the second configuration, y(x) = (λx/2)2 and we can write

where we have also replaced x by the radial coordinate r and κ is the spin-orbit coupling constant. Ω is the vector oscillator frequency whereas the pseudo-scalar oscillator frequency is ω. These parameter assignments are motivated by physical expectations as will be justified by the results obtained below. Without loss of generality, we can always choose V0 = 0. Therefore, the parameters in the Dirac wave operator matrix (A7a) are as follows:

Substituting these parameters in Eq. (A7a), results in the following symmetric three-term recursion relation for the expansion coefficients of the wave function

Dividing by 1/4 − 2(Ω44)[(ϵ + M)/λ] − ω44 and writing fn(ϵ) = f0(ϵ)Pn(ϵ), makes this a recursion relation for Pn(ϵ) with P0(ϵ) = 1. Comparing with that of the Meixner-Pollaczek polynomial (22) dictates that with

and

It is obvious that for positive energy where ϵM, these assignments violate reality since z becomes pure imaginary and |cos(θ)| > 1. Thus, we are forced to make the replacement z → iz and θ → iθ changing the trigonometric functions in Eq. (32) to hyperbolic and making the asymptotic wavefunction vanish since the oscillatory factor ei in Eq. (23) changes into a decaying factor e. All of this imply that there are no continuous energy scattering states but only discrete energy bound states. This, of course, is an expected result for the isotropic oscillator whose energy spectrum is confirmed using the spectrum formula of the Meixner-Pollaczek polynomial that gives

with Ω = 0, this is identical to the energy spectrum of the Dirac-oscillator problem. The corresponding m-th bound state wavefunction will be written in terms of the discrete version of the Meixner-Pollaczek polynomial as

4.2 The q = 1 Scenario of (A6b)

In this scenario, the vector potential is V(y) = y2a − 2(A + By) and the pseudo-scalar potential is W(y) = λya − 1(ρy + σ) such that ρ2 = 1/4. Moreover, the basis parameter ν is to be determined later by physical constraints whereas 2α = ν + 2 − a. There are two configurations in this scenario: one corresponds to a = 1 where y(x) = eλx with −∞ < x < +∞ and the other corresponds to a = 1/2 where y(x) = (λx/2)2.

For the first configuration, we can write

To make the vector potential vanish at infinity we can freely choose V0 = 0. Therefore, the parameters in the Dirac wave operator matrix (A7b) are as follows:

Substituting these parameters in Eq. (A7b), results in the following symmetric three-term recursion relation for the expansion coefficients of the wave function

Writing fn(ϵ) = f0(ϵ) Pn(ϵ), gives a recursion relation for Pn(ϵ) with P0(ϵ) = 1. We compare it to that of the orthonormal version of the three-parameter continuous dual Hahn polynomial that reads

where the continuous dual Hahn polynomial is defined as

with

Thus, we conclude that provided that

and , which requires that . The phase shift is obtained using the asymptotics formula of the continuous dual Hahn polynomial. It reads δμ(z) = arg[Γ(2iz)/Γ(ξ + iz) Γ(γ + iz)Γ(τ + iz)] giving

Now, the spectrum formula of the continuous dual Hahn polynomial reads , where n = 0, 1, 2, …, N and N is the largest integer less than or equal to −ξ. Consequently, we obtain the following relativistic energy spectrum formula

which is a quadratic equation to be solved for ϵn. With W0 = 0, it is identical to the energy spectrum of the spin-symmetric Dirac-Morse problem (i.e., with equal scalar and vector exponential potentials). Now, the continuous dual Hahn polynomial has a mix of continuous and discrete spectra for ξ < 0, then the following wavefunction represents the system with a mix of continuous energy ϵ and discrete energy ϵm

The corresponding orthogonality relation, which is valid for this case, reads as follows

For the second configuration, y(x) = (λx/2)2 and we can write

where we have replaced x by the radial coordinate r and κ is the spin-orbit coupling constant. To avoid quantum anomalies in the inverse square potential, we demand that its coupling strength be larger than the critical value of −1/4 (i.e., 8V12 > −1/4). Additionally, ω is the pseudo-scalar oscillator frequency and without loss of generality we can always choose V0 = 0. Therefore, the parameters in the Dirac wave operator matrix (A7b) are as follows:

Substituting these parameters in (A7b), results in the following symmetric three-term recursion relation for the expansion coefficients of the wavefunction

Rewriting fn(ϵ) = f0(ϵ)Pn(ϵ), makes this a recursion relation for Pn(ϵ) with P0(ϵ) = 1. Comparing to that of the continuous dual Hahn polynomial (37) makes such that

and z2 = −[(2V12)(ϵ + M) + (1/4) (κ + 1/2)2]. However, this implies that z is pure imaginary and the system has only discrete bound states. The spectrum formula of the continuous dual Hahn polynomial, , results in the following relativistic energy spectrum

which is to be solved for ϵn. With V1 = 0, this is identical to the energy spectrum of the Dirac-oscillator problem for an oscillator frequency ω2 = (1/2)λ2 and with ν = ±(κ + (1/2)) for ±κ > 0. The corresponding m-th bound state wavefunction will be written in terms of the dual Hahn polynomial , which is a discrete version of the continuous dual Hahn polynomial and defined as

where n, m = 0, 1, 2, …, N and either γ, τ > −1 or γ, τ < −N. Therefore, the mth bound state wavefunction is written as .

5 Solution in the Jacobi Basis

Let y(x) be a coordinate transformation such that −1 ⩽ y ⩽ +1. A complete set of square integrable functions as basis in the new configuration space with the dimensionless coordinate y has the following elements

where is the Jacobi polynomial of degree n in y and the normalization constant is chosen as

The real dimensionless parameters {μ, ν} are greater than −1 whereas {α, β} will be determined by square integrability and the tridiagonal requirement. Substituting Eq. (50) into Eq. (6) and using the differentiation chain rule d/dx = y′(d dy), we obtain

where the prime stands for the derivative with respect to x. It is required that in function space must be nearest neighbor to . The differential property of the Jacobi polynomial,

and its recursion relation,

show that this could be achieved if we impose the constraint that [(1 − y2)/y′]W(y) be a linear function in y. That is, [(1 − y2)/y′]W(y) = ρ(1 + y) − σ(1 − y) where ρ and σ are dimensionless potential parameters. Thus, we can rewrite Eq. (51) as follows

where

In the following two subsections, we consider the two physical scenarios found in Appendix B and corresponding to Eqs. (A16a) and (A16b), respectively.

5.1 The (p, q) = (0, 0) Scenario of (A16a)

In this scenario, the vector potential takes the form V(y) = (1 − y)2a − 1(1 + y)2b − 1(V0 + V1y), and the pseudo-scalar potential reads

where {V0, V1, W±} are real potential parameters. Moreover, the basis parameters are restricted to satisfy

2α = μ + 1 − a, and 2β = ν + 1 − b, where a and b are either (a, b) = (1/2, 1/2) or (a, b) = ( 0, 1/2). Additionally, the pseudo-scalar potential parameters become ρ = W/λ and σ = W+/λ.

For the first case, the solution of gives y(x) = sin(λx) with −π/2λ < x < π/2λ. The basis parameters become μ2 = (2W/λ − 1/2)2, ν2 = (2W+/λ − 1/2)2, 2α = μ + 1/2 and 2β = ν + 1/2. The potential functions read as follows

where we can always choose V0 = 0. The vector and scalar are potential boxes with sinusoidal bottom whereas the pseudo-scalar is a potential box with 1/x singularity of strength ±2W at the two edges of the box. This potential configuration was never reported in the literature. Its existence here is a demonstration of the unique advantage of the TRA over other methods for enlarging the class of exactly solvable potentials. Substituting these results back in (A16a), we obtain the following three-term recursion relation for the expansion coefficients

where Cn and Dn are defined in Eq. (52). Introducing

will transform Eq. (54) to the following recursion relation for Pn(ϵ)

Comparing Eq. (55) with Eq. (9) in Ref. [38], we conclude that

where

Some of the properties of this new polynomial (z−1; α, θ) were derived numerically in Ref. [38]. In contrast to the orthogonal polynomials of Sec. 3, the analytical properties of this new polynomial are not yet known. Thus, the properties of the corresponding physical system (such as the phase shift and energy spectrum that would have been determined from the asymptotics of the polynomial[38]) could not be given analytically or in a closed form. In the absence of these analytic properties, we give in Table 1 numerical results for the lowest part of the positive energy relativistic spectrum for a chosen set of values of the physical parameters. In Appendix C, we give the details of the procedure used in this calculation. The upper component of the spinor wavefunction is written as

where

The lower component of the spinor wavefunction can be easily obtained by calculating using Eq. (52) with ρ = W/λ and σ = W+/λ.

Table 1

The lowest part of the energy spectrum associated with the potential configuration (53) for various basis sizes. We used the procedure outlined in Appendix C and took the following values of the physical parameters: M = 1, λ = 1, V0 = 0, V1 = 5, W+ = −2, and W = 3.

.

For the second case where (a, b) = (0, 1/2), the solution of gives y(x) = 2(x/L)2 − 1 with 0 ≤ xL and . The basis parameters become μ2 = (LW − 1)2, ν2 = (LW+ − 1/2)2, 2α = μ + 1 and 2β = ν + 1/2, where we made the replacement . The solvable potential configuration reads

which are potential boxes with 1/x singularity at the edges of the box. Using Eq. (A16a), we write the three-term recursion relation associated with this relativistic system as follows

Following the same procedure as in the previous example, we define Qn(ϵ) = [A0/f0(ϵ)][fn(ϵ)/An], which transforms Eq. (57) to the following form

Thus, we conclude that

where

and cos(θ) = (ϵM − 2V0)/(ϵM + 2V1) (see Ref. [38] for details). Again, in the absence of analytic properties of the orthogonal polynomials Qn(ϵ), we give in Table 2 numerical results for the lowest part of the positive energy relativistic spectrum for a chosen set of values of the physical parameters. The spinor wavefunction components can be easily constructed using the same procedure followed in the previous problem.

Table 2

The lowest part of the energy spectrum associated with the potential configuration (56) for various basis sizes. We took the following values of the physical parameters: M = 1, L = 1, V0 = 5, V1 = −4, W+ = −2, and W = 3.

.
5.2 The (p, q) = (1, 0) Scenario of (A16b)

In this scenario, the vector potential is

and the pseudo-scalar potential is

where ρ = W/λ and σ = W+/λ. Moreover, the basis parameters are ν2 = (2W+/λ + b − 1)2, 2α = μ + 2 − a, and 2β = ν + 2 − b. The parameter μ is fixed later by physical constraints including the (finite) number of bound states. There are three physical configurations associated with this scenario. The first one corresponds to (a, b) = (1, 1/2) where and x ≥ 0. The second one corresponds to (a, b) = (1/2, 1/2) where y(x) = sin(λx) and −π/2λ < x < π/2λ. The third corresponds to (a, b) = (1, 0) where y(x) = 1 − 2e−λx and x ≥ 0.

For the first case and with , the solvable potential configuration reads

where we have also made the replacement . To force the vector potential to vanish at infinity, we choose V = 0. The basis parameters become ν2 = (W+/λ − 1/2)2, 2α = μ + 1 and 2β = ν + 1/2. Substituting these quantities in Eq. (A16b) and after somewhat lengthy manipulations, we obtain the following three-term recursion relation

where

Comparing this recursion relation with Eq. (12) in Ref. [38], gives , where is a new orthogonal polynomial defined in Ref. [38] with . Some of the interesting properties of this polynomial are discussed in the same Ref. [38]. For example, if σ is positive then this polynomial has only a continuous spectrum. However, if σ is negative then the spectrum is a mix of continuous scattering states and a finite number of discrete bound states. Moreover, the corresponding bound state energies are obtained from the following spectrum formula of the polynomial

where n = 0, 1, 2, …, N − 1 and N is the largest integer less than or equal to .

For the second configuration, (a, b) = (1/2, 1/2), which is equivalent to y(x) = sin(λx) with −π/2λ < x < π/2λ, and the potential functions read

where V > 0. It is interesting to note the difference between this case and the potential box in the first case of subsec. 5.1 above. The basis parameters become ν2 = (2W+/λ − 1/2)2, 2α = μ + 3/2 and 2β = ν + 1/2. Substitution in (A16b), we obtain a three-term recursion relation for this problem that resembles (60) above and reads as follows

where Tn(ϵ) = (A0/f0(ϵ))(fn(ϵ)/An) and again Bn = n + (μ + ν)/2 + 1. However, here

Thus, we can write . It should be noted that z2 < 0 indicating that the problem has only bound states with energies that are obtained from the spectrum formula (61). However, the spectrum here is infinite since the spectral terminating condition is satisfied for all integers.

For the last situation where (a, b) = (1, 0), we obtain y(x) = 1 − 2e−λx as solution of y′ = λ(1 − y) that satisfy y ∈ [−1, +1], where x ≥ 0. The basis parameters become ν2 = (2W+/λ − 1)2, 2α = μ + 1 and 2β = ν + 1. The solvable potentials now read

The above potentials are of the form of a generalized Hulthén potential with x being replaced {for} the radial coordinate r. To force the vector potential to vanish at infinity, we must choose V = 0. Substitution in Eq. (A16b) leads to the following three-term recursion relation for the expansion coefficients of the spinor wavefunction

where, again, Rn(ϵ) = [A0/f0(ϵ)][fn(ϵ)/An], and Bn = n + (μ + ν)/2 + 1. However,

which could be positive or negative depending on the sign of V0. Therefore, with and for negative σ the bound state energy spectrum is obtained from the spectrum formula (61). On the other hand, for positive σ the system has only continuum scattering states with the two-component wavefunction and where the scattering phase shift is obtained from the asymptotics of the polynomial Rn(ϵ), or equivalently , which is unfortunately not yet known analytically. Consequently, one needs to resort to numerical means.

6 Conclusion

In this article, we have discussed different exactly solvable potentials for the Dirac equation that have never been reported in the literature. However, we did not exhaust all possible solvable potentials in this manuscript. For example, we could have included a larger class of potentials by keeping V± ≠ 0 in the potential V(x) of Subsec. 5.1 provided that the basis parameters become energy dependent and chosen such that

Additionally, we could have also kept V+ ≠ 0 in the potential V(x) of subsec. 5.2 provided that the basis parameter ν is chosen such that

Moreover, we did not include the possibility that the basis is neither orthogonal nor tri-thogonal (i.e., the basis overlap matrix is not tridiagonal) but the Dirac wave operator is still tridiagonal. This is accomplished by the requirement that the matrix representation of the kinetic energy operator,

contains a counter term that cancels the non-tridiagonal .

We also hope that experts in orthogonal polynomials will soon derive the analytical properties of the two orthogonal polynomials mentioned in Sec. 5, which will allow us to write different properties associated with the physical system in closed form, e.g. the energy spectrum and phase shift.

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