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.

1 Introduction

We are concerned with the simplest quantum system: a single particle in a one-dimensional space. Its time evolution in the Schrödinger picture and in the coordinate representation is governed by the evolution equation:

$$\begin{aligned} \left[ i\hbar \frac{\partial }{\partial \,t}-{\mathsf {H}}(x)\right] \varPsi (x,t)=0. \end{aligned}$$

(1)

Solutions of the eigenvalue problem of the Hamiltonian

$$\begin{aligned} {\mathsf {H}}\,\psi _E(x)=E\,\psi _E(x) \end{aligned}$$

(2)

give the stationary state energies and wave functions. From here solution of (1) may be obtained as

$$\begin{aligned} \varPsi (x,t)={\textstyle \sum }\!\!\!\!\!\!\!\int _E\,{\mathcal {C}}(E)\, {\mathrm {e}}^{-iEt/\hbar }\,\psi _E(x)\,dE. \end{aligned}$$

(3)

We have chosen to describe a single fermion using the Lorentz-covariant quantum theory. The space-time is two-dimensional, and physical quantities may be described by scalars, pseudo-scalars and two vectors composed of time and space components. The relativistic dispersion relation for a free particle

$$\begin{aligned} {\mathsf {E}}^2=c^2{\mathsf {p}}^2+ m^2 c^4,\;\;\;{\mathsf {E}}={\mathsf {H}}_0,\;\;\; {\mathsf {p}}={\frac{\hbar }{i}}{\frac{\mathrm{d}}{\mathrm{{d}x}}}, \end{aligned}$$

(4)

implies that the free-particle Hamiltonian ${\mathsf {H}}_0$ is a Hermitian square root of the second-order differential operator in the rhs of this relation, i.e. it is the one-dimensional free Dirac operator [1]. By applying the classical procedure of Dirac, reduced to one dimension, we get

$$\begin{aligned} {\mathsf {H}}_0=c\pmb {\alpha }\,{\mathsf {p}}+\pmb {\beta }\,mc^2, \end{aligned}$$

(5)

with $\pmb {\alpha }\pmb {\beta }+\pmb {\beta }\pmb {\alpha }=0$, $\pmb {\alpha }^2=\pmb {\beta }^2=1$ and $x\in \langle -\infty ,\infty \rangle$¹. Since $\pmb {\alpha }$ and $\pmb {\beta }$ have to be Hermitian, their eigenvalues can only be $\pm {1}$. It is easy to see that ${\mathrm {Tr}}(\pmb {\alpha })=\mathrm {Tr}(\pmb {\beta })=0$ and, consequently, the dimension of these matrices has to be even. The Pauli matrices

$$\begin{aligned}&\pmb {\sigma }_0=\left( \begin{array}{cc}1&{}0\\ 0&{}1\end{array}\right) ,\;\; \pmb {\sigma }_1=\left( \begin{array}{cc}0&{}1\\ 1&{}0\end{array}\right) ,\;\;\\&\pmb {\sigma }_2=\left( \begin{array}{cc}0&{}-i\\ i&{}0\end{array}\right) ,\;\; \pmb {\sigma }_3=\left( \begin{array}{cc}1&{}0\\ 0&{}-1\end{array}\right) , \end{aligned}$$

form a complete set of $2\,\times \,2$ matrices. One can readily check that any two of three anti-commuting matrices fulfill the above conditions. Therefore two of them, say $\pmb {\sigma }_1$ and $\pmb {\sigma }_3$, can be set for $\pmb {\alpha }$ and $\pmb {\beta }$.

In realistic physical and chemical models, complicated multi-particle interactions are replaced by external fields—effective one-particle potentials which reflect the main features of these interactions in a simplified way. Models of a single particle in external fields may be applied to describe nearly all systems of chemical or physical interest. For example, a many-electron system may be reduced to a one-electron problem using the mean-field approach which results in Hartree–Fock or Dirac–Fock equations. The external potentials may have different transformation properties. In the “complete”, four-dimensional world the external potentials may transform as vectors, scalars, pseudo-scalars, pseudo-vectors, tensors. In our case we have a two-component vector, scalar and pseudo-scalar only. In the most general form, the one-dimensional Dirac operator is a linear combination of four Pauli matrices:

$$\begin{aligned} {\mathsf {H}}&=\pmb {\sigma }_0\,\mathsf {V}(x)+ c\,\pmb {\sigma }_1\left[ {\mathsf {p}}+ {\mathsf {U}}(x)\right] +c\,\pmb {\sigma }_2\,{\mathsf {P}}(x)\nonumber \\&+\pmb {\sigma }_3\left[ mc^2+{\mathsf {S}}(x)\right] \end{aligned}$$

(6)

where ${\mathsf {V}}$ and $c{\mathsf {U}}$ stand, respectively, for time and space components of the two-vector potential, $c\mathsf {P}$ for pseudo-scalar and $\mathsf {S}$—for scalar potential. Note that factor c in the definitions of the space component of the two-vector and of the pseudo-scalar potentials has been introduced for convenience and is related to the choice of units in which $\mathsf {U}$ and $\mathsf {P}$ are expressed.

The eigenvalue problem of the Dirac operator (6) reads

$$\begin{aligned} \mathsf {H}\,\psi (x)=E\,\psi (x), \end{aligned}$$

(7)

where $\psi$ is a two-component wave function. Using the gauge transformation

$$\begin{aligned} \psi (x)\,\rightarrow \,\exp \left( -\frac{i}{\hbar }\int \, \mathsf {U}(x)\mathrm{{d}}x\right) \psi (x), \end{aligned}$$

one can eliminate $\mathsf {U}(x)$. Then, without any loss of generality, we can assume that $\mathsf {U}(x)=0$.

Discussions of the eigenvalue problem of the one-dimensional Dirac operator (6) have appeared in the literature about half a century after the four-dimensional Dirac equation was formulated, mostly for didactic purposes. The discussion of the one-dimensional step potential has commonly been used as an illustration of the Klein paradox. A detailed early study has been given by Greiner [4]. General properties of the one-dimensional Dirac equation have been presented in a series of important works by Sukumar [5], Cooper et al. [6], Nogami, Toyama and coworkers [7,8,9,10]. According to a general observation by Wolfram [11]: “Even when the underlying rules for a system are extremely simple, the behavior of the system as a whole can be essentially arbitrarily rich”, many non-trivial aspects of the one-dimensional Dirac operator eigenvalue problem have been discovered with applications ranging from solid-state theory to nuclear and elementary particle physics. From the quantum-chemical perspective the most important are applications of the one-dimensional Dirac equation to the description of quantum systems whose spectrum has an energy gap and whose properties may be conveniently simulated by a properly taylored Dirac equation. To this category belong works on the edge states, on the Landau levels, on the theoretical modeling of graphene [12,13,14,15], also including external electromagnetic fields [16], and advanced studies on radially twisted carbon nanotubes [17, 18], specific calculations relevant to the theoretical simulation of one-dimensional graphene structures performed, among others, in [19, 20]. One-dimensional Dirac equation has also been used to the description of the $\pi$-electron systems in conjugated molecules [21].

During the last two decades, the number of papers concerned with one-dimensional Dirac eigenvalue problem has grown up exponentially. The subject covers many areas of modern physics, as, for example, statistical properties of spectra [22, 23], properties of nonlinear equations [24] and studies on the Cornell potential describing quark interactions [25,26,27]. A great deal of work has been done on the one-dimensional hydrogen atom—an issue apparently simple, but in fact, containing many mathematical subtleties and still, to a certain extent controversial. Already the choice of the potential is not obvious. The solution of the Poisson equation in one dimension for the point charge is proportional to |x|. Thus, some authors describe the 1D hydrogen atom using this potential [28,29,30] or its screened form [31]. In some other works, the singular 1D limit a/|x| of the 3D Coulomb potential a/r or its regularized forms is used instead [32, 33]. Different kinds of potentials lead to different forms of equations and, consequently, to different, sometimes unexpected, spectral properties. For example, as shown in [34, 35] the pseudo-scalar Coulomb problem has no bound states, also in 2D and 3D cases [34]. On the other hand de Castro, in a very interesting paper [36], demonstrated that there exists an infinite sequence of bound states if the problem is formulated in a slightly different way. Spin and pseudo-spin symmetries of the 1D Dirac equation with Coulomb potential were studied in [37]. A unified treatment of Coulomb and harmonic potentials in D dimensions, including $D=1$, have been discussed by Lévai et al. [38].

The one-dimensional Dirac equation is a nice playground for designing analytical solutions in some non-trivial environments. Here are some examples: Bound states in double-well potentials [39], pseudo-scalar potential barrier [40], construction of transparent potentials [41, 42], quadratic plus inversely quadratic potential [43], Kratzer potential [44], Wood-Saxon potential and effective mass problem [45], hyperbolic tangent potential [46] and Dirac–Moshinsky oscillator [47], just to mention several examples. Among innovative, non-standard, applications one should mention a work by Correa and Jakubsky on the description of optical systems, in the coupled mode theory of the Bragg gratings using 1D Dirac equation with a non-Hermitian Hamiltonian [48]. For some potentials the 1D Dirac equation is also quasi-exactly solvable. For example, in Ref. [49] quasi-exact solutions for the hyperbolic secant potential have been found.

The one-dimensional Dirac equation is reducible to a Schrödinger form in which the original Dirac potential generates a two- or three-term effective potential. The integrability of this Schrödinger equation implies that the Dirac equation is also solvable and its solutions are directly expressible in terms of the solutions of the Schrödinger equation. During the last decade, a large family of Schrödinger equations, reducible to different forms of the Heun equation, have been solved analytically (see, for example, [50,51,52]). Consequently, also the set of solvable Dirac equations is extended. An example, the analytical solution for the inverse square root potential, has recently been presented [53].

Due to its simplicity, the one-dimensional Dirac equation is a practical and convenient tool to study the interplay between different kinds of potentials. Among many works devoted to this subject, let us mention some representative examples. Thus, the dependence of the asymptotic behavior of the effective confining potentials on the relative strength of vector and scalar contributions was studied in [54], a joint treatment of mixed vector–scalar potentials in [55, 56] and mixed vector–scalar–pseudo-scalar potentials in [57].

In this paper we analyze some general properties of the one-dimensional Dirac operator. In the consecutive sections, we discuss the structure of the energy spectrum, the non-relativistic limit of the equation and, formally related to it, spin and pseudo-spin symmetries. Then, the large-component equation and the Pauli approximation, important in analyzing the asymptotic behavior of the eigen-problem at the non-relativistic limit, are presented. Next, the position dependent mass is discussed in a general context of the one-dimensional relativistic model. Finally, in the last two sections we discuss the reduction of the Dirac equation to the Schrödinger form and the equations describing massless fermions.

The paper is dedicated to our old friend, Professor Ramon Carbó-Dorca on the occasion of his 80th birthday.

2 General properties of the spectrum

The energy spectrum of the one-dimensional free Dirac operator, consists of two continua. The positive one spreads from the rest energy of the particle, $mc^2$, to $+\infty$. It corresponds to the non-relativistic energies of a free particle. The negative continuum, spreading from $-mc^2$ to $-\infty$, has no physical meaning. But in the charge-conjugate equation the negative energy states correspond to positrons with positive energies. The two continua in the spectrum of the Dirac operator are referred to as its essential spectrum. The essential spectrum of a Hermitian operator is a subset of its complete spectrum; its complement is the discrete spectrum, i.e. the set of isolated eigenvalues of finite multiplicity. For most of potentials of physical significance the discrete energies of the bound states are located in the energy gap between $-mc^2$ and $mc^2$. For some specific potentials resonances embedded in the continuum, described by the wave functions containing both localized and continuum-type contributions, and corresponding to auto-ionizing states, are also present [58, 59].

Many symmetries, present in three-dimensional space, are absent in one dimension. One of the consequences is that there is no degeneracy of energy levels in one dimension [7]. Since angular momentum does not exist in one dimension, spin effects, in particular spin-orbit interaction, are absent in one-dimensional Dirac equation. This simplifies the mathematical structure of the Dirac operator eigenvalue equation, compared to the three-dimensional case. Therefore, in one dimension some detailed studies on the interplay between different kinds of interactions are easier and more transparent than in the three-dimensional case. Let us note that the vector potentials couple to the electric charges. Then, the coupling is different for particles and for antiparticles and spectrum is not symmetric with respect to change of the energy sign. The scalar interactions couple potentials to the mass, the same for particles and for antiparticles. As a consequence, the energy spectrum is symmetric: to each bound state of a particle with energy E, there exists an antiparticle counterpart with energy $-E$ [7]. For the same reasons the effect of confinement due to vector and scalar potentials is different. In the case of vector coupling if $\mathsf {V}(x)\rightarrow \infty$ for $|x|\rightarrow \infty$, bound states do not exist—all states are metastable. For scalar coupling, similarly as in the non-relativistic models, we have an infinite series of bound states in such cases [9].

3 Non-relativistic limit

The eigenvalue equation (7), with the gauge leading to $\mathsf {U}(x)=0$, may be written as

$$\begin{aligned} \left( \begin{array}{cc} \mathsf {V}^+-E^-, &{} c\,\pi ^\dagger \\ c\,\pi , &{} \mathsf {V}^--E^+ \end{array}\right) \left( \begin{array}{c} \psi _\mathrm {L}\\ \psi _\mathrm {S} \end{array}\right) =0, \end{aligned}$$

(8)

where

$$\begin{aligned} E^\pm =E\,\pm \,mc^2\;\;\;\; \mathsf {V}^\pm =\mathsf {V}\pm \mathsf {S},\;\;\;\; \pi =\mathsf {p}+i\mathsf {P} \end{aligned}$$

and

$$\begin{aligned} \psi (x)=\left( \begin{array}{c} \psi _\mathrm {L}(x)\\ \psi _\mathrm {S}(x) \end{array}\right) . \end{aligned}$$

Two components of the wave function, $\psi _\mathrm {L}$ and $\psi _\mathrm {S}$, are referred to, respectively, as large and small components—the names restricted to the present representation of the Dirac operator in the two-dimensional spinor space, known as the Pauli representation.

In order to get the non-relativistic limit of Eq. (8) it is convenient to introduce the “non-relativistic energy scale”, i.e. to shift the energy by the rest energy of the particle. Then, Eq. (8) becomes

$$\begin{aligned} \left( \begin{array}{cc} \mathsf {V}^+-\mathcal {E}, &{} c\,\pi ^\dagger \\ c\,\pi , &{} \mathsf {V}^--2mc^2-\mathcal {E} \end{array}\right) \left( \begin{array}{c} \psi _\mathrm {L}\\ \psi _\mathrm {S} \end{array}\right) =0, \end{aligned}$$

(9)

where $\mathcal {E}=E^-$. After dividing the second of the two equations by c, introducing $c\,\psi _\mathrm {S}$ instead of $\psi _\mathrm {S}$ and setting $(\mathsf {V}^--\mathcal {E})/(2mc^2)=0$, Eq. (9) transforms to

$$\begin{aligned} \left( \begin{array}{cc} \mathsf {V}^+-\mathcal {E}, &{} \pi ^\dagger \\ \pi , &{} -2m \end{array}\right) \left( \begin{array}{c} \psi _\mathrm {L}\\ c\,\psi _\mathrm {S} \end{array}\right) =0, \end{aligned}$$

(10)

i.e. to the non-relativistic limit defined in the same Hilbert space as the original Dirac-operator eigenvalue problem. This equation, known as the one-dimensional Lévy-Leblond equation, transforms according to a spinor representation of the Galilei group. The elimination of $c\,\psi _\mathrm {S}$ leads to the Schrödinger form of the non-relativistic limit:

$$\begin{aligned} \left[ \frac{\pi ^\dagger \pi }{2m}+\mathsf {V}^+\right] \psi _\mathrm {L} =\mathcal {E}\psi _\mathrm {L}. \end{aligned}$$

(11)

As one can see, in the non-relativistic limit $\mathsf {V}$ and $\mathsf {S}$ are combined to form the non-relativistic “scalar” potential $\mathsf {V}^+$. The relativistic pseudo-scalar potential $\mathsf {P}$ enters the kinetic energy operator in a similar way as a single component of the non-relativistic “vector potential”:

$$\begin{aligned} \pi ^\dagger \pi =(\mathsf {p}-i\mathsf {P})(\mathsf {p}+i\mathsf {P})= -\hbar ^2\frac{d^2}{dx^2}+\hbar \frac{d\mathsf {P}}{dx}+\mathsf {P}^2. \end{aligned}$$

4 Spin and pseudo-spin symmetries

A straightforward reduction of the eigenvalue equation (8) to a Schrödinger-like equation is obtained in the case of spin symmetry, or pseudo-spin symmetry, i.e. when, respectively, $\mathsf {V}^-=v^-=\mathrm {const}$ or $\mathsf {V}^+=v^+=\mathrm {const}$. In both cases one of the components of $\psi (x)$ in Eq. (8) may be eliminated and the equations for the second component read, respectively:

$$\begin{aligned}&\left[ \frac{\pi ^\dagger \,\pi }{2\mathfrak {M}_v^-}+ \mathsf {V}^+\right] \psi _\mathrm {L} =E^-\,\psi _\mathrm {L},\;\;\;\;\nonumber \\&\hbox {and}\;\;\;\; \left[ \frac{\pi \,\pi ^\dagger }{2\mathfrak {M}_v^+}+ \mathsf {V}^-\right] \psi _\mathrm {S} =E^+\,\psi _\mathrm {S}, \end{aligned}$$

(12)

where

$$\begin{aligned} \mathfrak {M}_v^\mp =m \left( \frac{E^\mp -v^\mp }{2mc^2}\,\pm \,1\right) . \end{aligned}$$

Note that in the limit of small external potentials, the dominant component is $\psi _\mathrm {L}$ for spin symmetry and $\psi _\mathrm {S}$ for pseudo-spin symmetry. In both cases, if for a given potential analytic solutions of the Schrödinger equation are known, then the corresponding solutions of the one-dimensional Dirac equation may be easily obtained by a direct substitution of the appropriate parameters. A highly instructive note on this subject, with wealth of references, has been presented in a short comment by Castro and de Castro [26].

It has hardly been noticed that the reduction  of the Dirac eigenvalue problem to the Schrödinger-like one by suppressing the coupling between components of the Dirac wave function has originally been done by Moore nearly half a century ago. The resulting equation

$$\begin{aligned} \mathsf {H}_\mathrm {d}\,\psi _\mathrm {L}=\mathcal {E}\psi _\mathrm {L}, \end{aligned}$$

(13)

where

$$\begin{aligned} \mathsf {H}_\mathrm {d}=\frac{\pi ^\dagger \,\pi }{2\,\mathfrak {M}}+ \mathsf {V},\;\;\;\;\mathfrak {M}=m\left( 1+\frac{\mathcal {E}}{2mc^2}\right) \end{aligned}$$

is known as the direct Dirac equation [60].²

5 Large-component equation

After some simple algebra, the separation of Eq. (8) yields

$$\begin{aligned} \left[ \frac{\pi ^\dagger \,\mathsf {R}\,\pi }{2\mathfrak {M}}+ \mathsf {V}^+\right] \psi _\mathrm {L}=\mathcal {E}\psi _\mathrm {L}, \end{aligned}$$

(14)

where

$$\begin{aligned} \mathsf {R}=\frac{1}{1-\mathsf {w}},\;\;\;\;\; \mathsf {w}=\frac{\mathsf {V}^-}{2\mathfrak {M}\,c^2} \end{aligned}$$

(15)

and

$$\begin{aligned} \psi _\mathrm {S}=\frac{\mathsf {R}\,\pi }{2\mathfrak {M}\,c}\, \psi _\mathrm {L}. \end{aligned}$$

(16)

The pair of Eqs. (14) and (16) is equivalent to the original Dirac equation. But, Eq. (14) itself is not an eigenvalues problem—parameter $\mathfrak {M}$ depends on $\mathcal {E}$ and, consequently, functions corresponding to different $\mathcal {E}$ are not orthogonal. The orthonormality condition involves both $\psi _\mathrm {L}$ and $\psi _\mathrm {S}$ and reads

$$\begin{aligned} \int _{-\infty }^\infty \,{\psi ^{(m)}}^\dagger (x)\,\psi ^{(n)}(x)\,\mathrm{{d}}x= \delta _{mn}, \end{aligned}$$

where $\psi ^{(m)}$ and $\psi ^{(n)}$ are two-component wave functions (8) corresponding, respectively, to energies $\mathcal {E}_m$ and $\mathcal {E}_n$.

Equation (14) may be rewritten as

$$\begin{aligned} \left[ \mathsf {H}_0+\mathsf {H}_\mathrm {D} +\mathsf {H}_\mathrm {mv}\right] \psi _\mathrm {L}=\mathcal {E}\psi _\mathrm {L}, \end{aligned}$$

(17)

where

$$\begin{aligned} \mathsf {H}_0=\frac{\pi ^\dagger \,\pi }{2\,m}+ \mathsf {V}^+, \end{aligned}$$

and

$$\begin{aligned} \mathsf {H}_\mathrm {D}=\frac{\mathsf {R}}{2\mathfrak {M}}\, \frac{\mathsf {p}(\mathsf {V}^-)\pi }{2m\,c^2},\;\;\;\; \mathsf {H}_\mathrm {mv}= -\frac{(\mathsf {V}^+-\mathcal {E})(\mathsf {V}^--\mathcal {E})}{2mc^2}; \end{aligned}$$

$\mathsf {H}_\mathrm {D}$ and $\mathsf {H}_\mathrm {mv}$ may be recognized as the exact form of the one-dimensional Darwin correction and the mass-velocity term. The standard Pauli approximation results from Eq. (17) by the replacement of $\mathsf {R}$ and $\mathfrak {M}$ by, respectively, 1 and m.

5.1 Position-dependent effective mass

Equation (14) may also be interpreted as the Schrödinger equation with a position-dependent effective mass, commonly used in solid-state physics [61]. If we define the effective mass as

$$\begin{aligned} m^*(x)=\frac{\mathfrak {M}}{\mathsf {R}(x)}= \mathfrak {M}\,\left( 1-\frac{\mathsf {V}^-}{\mathfrak {M}\,c^2} \right) \end{aligned}$$

(18)

then, Eq. (14) yields

$$\begin{aligned} \left[ \pi ^\dagger \,\frac{1}{2\,m^*(x)}\,\pi + \mathsf {V}^+\right] \Phi (x)=\mathcal {E}\Phi (x), \end{aligned}$$

(19)

where $\Phi (x)$ is the wave function which, effectively, contains relativistic effects carried implicitly by the effective mass $m^*$. The model is formally valid if

$$\begin{aligned} F(x)\,\equiv \,\frac{1}{m^*(x)}\frac{d\,\Phi (x)}{dx} \end{aligned}$$

is continuous [61].

6 Reduction to a Schrödinger form

In some cases the reduction of the one-dimensional Dirac equation to the Schrödinger form is associated with the factorization of the Hamiltonian to two mutually Hermite conjugate operators which, in turn, is the first step in an analysis of super-symmetry (SUSY) properties of the equation. Studies on super-symmetry of the one-dimensional Dirac equation were initiated by fundamental works by Sukumar [5], Cooper et al. [6], Nogami and Toyama [8]. Basic ideas of SUSY quantum mechanics, its relations to exact solubility of quantum problems and a list of exactly solvable potentials have been given by Dutt at al. [62]. In the next subsections two special cases are considered.

6.1 Constant $\mathsf {P}$ and $\mathsf {V}(x)\sim \mathsf {S}(x)$

Let $\mathsf {P}(x)=\mathsf {P}_0$, where $\mathsf {P}_0$ is x-independent, and

$$\begin{aligned} \mathsf {V}^+(x)=q\,\mathsf {W}(x),\;\;\;\;\; \mathsf {V}^-(x)=-r\,\mathsf {W}(x), \end{aligned}$$

(20)

where $q,r\,\ne \,0$ are independent of x. If $q=0$ or $r=0$, then we have special cases of spin or pseudo-spin symmetries, considered in Sect.  4. Equation (8) may be rewritten as

$$\begin{aligned} \left( \begin{array}{cc} c\,(\mathsf {p}+i\,\mathsf {P}_0), &{} -r\,\mathsf {W}(x)-E^+\\[5pt] q\,\mathsf {W}(x)-E^-, &{} c\,(\mathsf {p}-i\,\mathsf {P}_0) \end{array} \right) \left( \begin{array}{c} \psi _\mathrm {L}(x)\\[5pt] \psi _\mathrm {S}(x) \end{array} \right) =0. \end{aligned}$$

(21)

By a similarity transformation:

$$\begin{aligned}&\mathsf {T}=\frac{1}{\sqrt{2qr}}\left( \begin{array}{cc} \sqrt{qr}, &{} i\,r\\[3pt] i\,q, &{} \sqrt{qr}\\ \end{array}\right) ,\;\;\;\nonumber \\&\mathsf {T}^{-1}=\frac{1}{\sqrt{2qr}}\left( \begin{array}{cc} \sqrt{qr}, &{} -i\,r\\[3pt] -i\,q, &{} \sqrt{qr} \end{array}\right) , \end{aligned}$$

(22)

Eq. (21) becomes

$$\begin{aligned} \left( \begin{array}{cc} c\left( \mathsf {p}+i\,\mathsf {Q}(x)\right) , &{} r\,g^-\\[5pt] q\,g^+, &{} c\left( \mathsf {p}-i\,\mathsf {Q}(x)\right) \end{array} \right) \left( \begin{array}{c} \varphi _1(x)\\[5pt] \varphi _2(x) \end{array} \right) =0, \end{aligned}$$

(23)

where

$$\begin{aligned} \mathsf {Q}(x)=\mathsf {Y}(x)+\mathfrak {p}_\mathsf {V}, \end{aligned}$$

(24)

with

$$\begin{aligned} \mathsf {Y}(x)=\frac{\sqrt{qr}}{c}\,\mathsf {W}(x),\;\;\;\;\;\; \mathfrak {p}_\mathsf {V}=\frac{q\,E^+-r\,E^-}{2c\,\sqrt{qr}} \end{aligned}$$

(25)

and

$$\begin{aligned} \varphi (x)&=\left( \begin{array}{c}\varphi _1(x)\\ \varphi _2(x)\end{array}\right) =\mathsf {T}\,\psi (x), \;\;\;\;\;\;\;\\ g^\pm&=-\frac{1}{\sqrt{qr}}\left( \frac{q\,E^++r\,E^-}{2\sqrt{qr}} \pm \,c\,\mathsf {P}_0\right) . \end{aligned}$$

By introducing

$$\begin{aligned} A^\pm (x)=i\,\left( -\hbar \frac{d}{dx}\pm \,\mathsf {Q}(x) \right) , \end{aligned}$$

(26)

Eq. (23) implies

$$\begin{aligned}&\left( \mathsf {A}^-\,\mathsf {A}^++\mathfrak {p}_\mathsf {V}^2\right) \,\varphi _1(x)= \mathfrak {E}_\mathsf {V}\,\varphi _1(x),\;\;\;\;\nonumber \\&\left( \mathsf {A}^+\,\mathsf {A}^-+\mathfrak {p}_\mathsf {V}^2\right) \,\varphi _2(x)= \mathfrak {E}_\mathsf {V}\,\varphi _2(x), \end{aligned}$$

(27)

$\mathfrak {p}_\mathsf {V}^2/2m$ is the factorization energy [6] and

$$\begin{aligned} \mathfrak {E}_\mathsf {V}=\frac{E^+\,E^-}{c^2}-\mathsf {P}_0^2= 2\,\mathfrak {M}\,\mathcal {E}-P_0^2. \end{aligned}$$

(28)

Eq. (27) may be rewritten in the Schrödinger form:

$$\begin{aligned} \left( -\hbar ^2\frac{d^2}{dx^2}+\mathsf {Y}^2+ 2\mathfrak {p}_\mathsf {V}\mathsf {Y}\,\mp \,\hbar \frac{d\,\mathsf {Y}}{dx} \right) \varphi _{1,2}=\mathfrak {E}_\mathsf {V}\varphi _{1,2}. \end{aligned}$$

(29)

Hamiltonian in Eq. (29) depends on parameter E. For a fixed value of E, the Hamiltonian has a complete spectrum of eigenvalues. If an eigenvalue is equal to the parameter E, then it is also equal to the corresponding eigenvalue of the Dirac Hamiltonian. The tuning of an eigenvalue to the value of the parameter is straightforward if the equation is solved analytically. In case of numerical solutions, an iterative loop in which the two numbers are adjusted to each other is necessary. In any case, at most one eigenvalue may be equal to a fixed parameter.

According to (20)

$$\begin{aligned} \frac{\mathsf {S}^2-\mathsf {V}^2}{\mathsf {W}^2}=qr. \end{aligned}$$

If $qr>0$, i.e. if the scalar part dominates in the total potential, then the eigenvalue problem (29) is Hermitian. If $qr<0$, then the Hamiltonian is complex, i.e. it is non-Hermitian. But, if $\mathsf {W}(x)=\mathsf {W}^*(-x)$ then Eq. (29) is $\mathcal {P\,T}-$symmetric.³

Two cases deserve some particular attention:

• 1. $q=r=1$. In this case $\mathsf {V}=0$, i.e. $\mathsf {V}^+=-\mathsf {V}^-=\mathsf {S}$, $\;\mathsf {Y}=\mathsf {W}/c$ and $\;\mathfrak {p}_\mathsf {V}=mc$. Equation (29) simplifies to a Hermitian eigenvalue problem with E-independent Hamiltonian: It is interesting to note that the Hamiltonian does not depend on E only if $q=r$.

• 2. $q=-r=1$. In this case $\mathsf {S}=0$, i.e. $\mathsf {V}^+=\mathsf {V}^-=\mathsf {V}$, $\;\mathsf {Y}=i\,\mathsf {W}/c$ and $\;\mathfrak {p}_\mathsf {V}=E/c$. The Hamiltonian depends on E and in general is non-Hermitian, but if $\mathsf {W}(x)=\mathsf {W}^*(-x)$ then Eq. (29) is $\mathcal {P\,T}$-symmetric:

  $$\begin{aligned}&\left[ -\hbar ^2\frac{d^2}{dx^2}-\left( \frac{\mathsf {W}}{c}\right) ^2 +\frac{2\,E}{c}\left( \frac{\mathsf {W}}{c}\right) \,\right. \\&\left. \qquad \mp \,i\hbar \frac{d}{dx}\left( \frac{\mathsf {W}}{c}\right) \right] \varphi _{1,2}=\mathfrak {E}_\mathsf {V}\varphi _{1,2}. \end{aligned}$$

6.2 Constant $\mathsf {V}^+$ and $\mathsf {V}^-$

If $\mathsf {V}^+(x)=v^+$, $\mathsf {V}^-(x)=v^-$, where $v^+,v^-$ are x independent, then Eq. (8) may be rewritten as

$$\begin{aligned} \left( \begin{array}{cc} v^+-E^-, &{} c\,\left( \mathsf {p}-i\widetilde{\mathsf {P}}(x)\right) \\[5pt] c\,\left( \mathsf {p}+i\widetilde{\mathsf {P}}(x)\right) , &{} v^--E^+ \end{array} \right) \left( \begin{array}{c} \psi _\mathrm {L}(x)\\[5pt] \psi _\mathrm {S}(x) \end{array} \right) =0, \end{aligned}$$

(30)

with $\widetilde{\mathsf {P}}(x)=\mathsf {P}(x)+\mathsf {P}_0$, where $\mathsf {P}_0$ is a constant background potential [36, 64]. Equation (30) may be easily decoupled:

$$\begin{aligned} \psi _\mathrm {s,l}(x)=\frac{c\, \left( \mathsf {p}{\pm }i\widetilde{\mathsf {P}}(x)\right) }{E^\pm -v^\mp }\,\psi _\mathrm {l,s}(x) \end{aligned}$$

(31)

and

$$\begin{aligned}&\left( -\hbar ^2\frac{d^2}{dx^2}+\mathsf {P}^2(x)+ 2\mathsf {P}_0\,\mathsf {P}(x)\pm \hbar \frac{d\mathsf {P}}{dx}\right) \nonumber \\&\qquad \qquad \psi _\mathrm {l,s}(x)= \mathfrak {E}_\mathsf {P}\,\psi _\mathrm {l,s}(x) \end{aligned}$$

(32)

where

$$\begin{aligned} \mathfrak {E}_\mathsf {P}=\left( \mathcal {E}-v^+\right) \left( \frac{\mathcal {E}-v^-}{c^2}+2m\right) -\mathsf {P}_0^2. \end{aligned}$$

(33)

Note, that if we set $\mathsf {P}=\mathsf {W}/c$ and $\mathsf {P}_0=mc$, then the Hamiltonians in Eq. (32) are the same as in Eq. (29), in the case of the scalar potential. By introducing

$$\begin{aligned} A^\pm (x)=i\, \left( -\hbar \frac{d}{dx}\pm \,\widetilde{\mathsf {P}}(x)\right) \end{aligned}$$

we can rewrite Eq. (32) as a SUSY pair of equations:

$$\begin{aligned} \mathsf {A}^+\,\mathsf {A}^-\,\psi _\mathrm {L}(x)= \mathfrak {E}_\mathsf {P}\,\psi _\mathrm {S}(x),\;\;\;\; \text{ and }\;\;\;\; \mathsf {A}^-\,\mathsf {A}^+\,\psi _\mathrm {S}(x)= \mathfrak {E}_\mathsf {P}\,\psi _\mathrm {S}(x) \end{aligned}$$

7 Massless fermions and shift of potentials

The time component of the vector potential, $\mathsf {V}(x)$, according to the minimal coupling principle, enters the Dirac equation in the same way as energy. Therefore, a shift by a constant, $\mathsf {V}\,\rightarrow \,\mathsf {V}+e$, is equivalent to shifting the energy scale by e. Consequently, under this transformation the energy spectrum is shifted by e and the form of the eigenfunctions does not change.

The scalar potential $\mathsf {S}$, under the Lorentz transformation, behaves like a scalar and is included to the Dirac equation in the same way as the particle mass. In the non-relativistic limit $\mathsf {S}$ and $\mathsf {V}$ are non-distinguishable (the time component of the vector potential, under the Galilei transformation also behaves like a scalar) and combine to form a single potential. But, in the Dirac equation the behavior of $\mathsf {S}$ and $\mathsf {V}$ is different. In particular, neither energies nor eigenfunctions are invariant with respect to a shift of $\mathsf {S}$. In general, $\mathsf {S}$ is like a position-dependent mass. By setting $\mathsf {S}=mc^2$ we get Dirac equation describing massless fermions, quasi-particles playing an important role in the modeling of graphene [14].

Finally, the pseudo-scalar potential $\mathsf {P}$ adds to the momentum operator. Its influence on the spectrum of the Dirac operator is complicated. In particular, by adding a constant to a pseudo-scalar potential one may change the spectrum of an operator which does not support bound states to another one, with infinite number of bound states, as it was demonstrated for an inversely linear [64] and for a screened Coulomb [65] potentials.

The 1D Dirac equation for massless fermions has a simple, symmetric, form

$$\begin{aligned} \left( \begin{array}{cc} \mathsf {V}-E, &{} c\,\pi ^\dagger \\ c\,\pi , &{} \mathsf {V}-E \end{array}\right) \left( \begin{array}{c} \psi _\mathrm {a}\\ \psi _\mathrm {b} \end{array}\right) =0. \end{aligned}$$

(34)

This means that particle and antiparticle spectra are the same. If $\mathsf {P}=0$, i.e. we have pure vector potential, the equation can be decoupled by transforming to the Weyl representation. By introducing $\phi _{1,2}=\left( \psi _\mathrm {a}\pm \psi _\mathrm {b}\right) /\sqrt{2}$ we get

$$\begin{aligned} \frac{i}{\phi _1}\frac{d\phi _1}{dx}= & {} \frac{\mathsf {V}-E}{\hbar \,c}\\ -\frac{i}{\phi _2}\frac{d\phi _2}{dx}= & {} \frac{\mathsf {V}-E}{\hbar \,c}. \end{aligned}$$

After the integration

$$\begin{aligned} \phi _{1,2}\,\sim \,\exp \left[ \pm \,i\int \frac{(\mathsf {V}-E)}{\hbar \,c}dx \right] , \end{aligned}$$

where E is arbitrary. Then, the wave functions are not square integrable and the spectrum is continuous.

The introduction of a constant pseudo-scalar contribution, $\mathsf {P}_0$, changes this picture. The 1D Dirac equation expressed in the Weyl representation

$$\begin{aligned} \left( \begin{array}{cc} c\,\mathsf {p}+\mathsf {V}-E, &{} ic\,\mathsf {P}_0\\ -ic\,\mathsf {P}_0, &{} -c\,\mathsf {p}+\mathsf {V}-E \end{array}\right) \left( \begin{array}{c} \phi _\mathrm {1}\\ \phi _\mathrm {2} \end{array}\right) =0, \end{aligned}$$

(35)

may be written in the second-order form

$$\begin{aligned} \left[ c^2\,\mathsf {p}^2-(\mathsf {V}-E)^2\pm (\mathsf {p}\mathsf {V}) +c^2\,\mathsf {P}_0^2\right] \phi _{1,2}=0, \end{aligned}$$

(36)

with

$$\begin{aligned} \phi _{2,1}= \frac{\pm \,c\,\mathsf {p}+\mathsf {V}-E}{\mp \,ic\,\mathsf {P}_0}\, \phi _{1,2}. \end{aligned}$$

Eq. (36) may easily be transformed to an eigenvalue problem with a complex and energy-dependent effective potential.

In the case of a pure pseudo-scalar potential, i.e. $\mathsf {V}=0$, Eq. (34) yields

$$\begin{aligned} \left[ c^2(\mathsf {p}\mp \,i\mathsf {P})(\mathsf {p}\pm \,i\mathsf {P})- E^2\right] \phi _\mathrm {a,b}=0. \end{aligned}$$

(37)

The Hamiltonian in the corresponding eigenvalue problem is Hermitian. The symmetry of particle–antiparticle spectrum is reflected in a symmetry of the wave functions: the replacement $(+E\leftrightarrow -E)$ corresponds to the replacement of the components $(\phi _1\leftrightarrow \phi _2)$ in the two-component spinors.

8 Final remarks

In all cases considered in Sects. 6 and 7 the Schrödinger forms of the Dirac equation, i.e. Eqs. (29) and (30), after the substitution $x/\hbar \,\rightarrow \,x$, may be written as

$$\begin{aligned} -\frac{\mathrm{d}^2\chi }{\mathrm{d}x^2}+ \mathsf {V}_\mathrm {eff}(x)\,\chi (x)=\mathfrak {E}\,\chi (x), \end{aligned}$$

(38)

where

$$\begin{aligned} \mathsf {V}_\mathrm {eff}(x)=\mathsf {Y}^2(x)+2b\, \mathsf {Y}(x)\mp \frac{\mathrm{d}\mathsf {Y}}{\mathrm{d}x}, \end{aligned}$$

and, depending on the case, $\mathsf {Y}(x)=\sqrt{qr}\,\mathsf {W}/c$ with $b=\mathfrak {p}_\mathsf {V}$ or $\mathsf {Y}(x)=\mathsf {P}(x)$ with $b=\mathsf {P}_0$ and $\mathfrak {E}$ stands for $\mathfrak {E}_\mathsf {V}$ or $\mathfrak {E}_\mathsf {P}$. The resulting Hamiltonian is Hermitian, unless $qr\,<\,0$—then, for $\mathsf {W}(x)=\mathsf {W}^*(-x)$, it is $\mathcal {P\,T}$-symmetric. Let us note that if we allow for imaginary potentials, then Eq. (38) corresponds to the one-dimensional Dirac equation with scalar potential

$$\begin{aligned} \mathsf {S}(x)=c\,\mathsf {Y}(x)+b. \end{aligned}$$

The effective potential may be factorized and represented in a SUSY form. Many solvable representations of this potential have already been discussed in detail by authors mentioned in the Introduction. For a large class of potentials the ground-state energies and wave functions can be obtained by a direct integration of the equation.

Many new potentials, particularly the ones linked with different members of the Heun family of equations remain to be explored. Also the problem of metastable states has scarcely been discussed [66] and very little was done on Hamiltonians which are neither Hermitian nor $\mathcal {P\,T}$-symmetric. We conclude that the subject “One-dimensional Dirac equation” is far from being exhausted.