Dirac systems with magnetic field and position dependent mass: Darboux transformations and equivalence with generalized Dirac oscillators
aa r X i v : . [ qu a n t - ph ] F e b DIRAC SYSTEMS WITH MAGNETIC FIELD ANDPOSITION-DEPENDENT MASS: DARBOUX TRANSFORMATIONSAND EQUIVALENCE WITH GENERALIZED DIRAC OSCILLATORS
Axel Schulze-Halberg † and Pinaki Roy ‡ , ∗ † Department of Mathematics and Actuarial Science and Department of Physics, IndianaUniversity Northwest, 3400 Broadway, Gary IN 46408, USA, E-mail: [email protected] ‡ Atomic Molecular and Optical Physics Research Group, Advanced Institute of Materi-als Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam ∗ Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam,E-mail: [email protected]
Abstract
We construct a Darboux transformation for a class of two-dimensional Dirac systems at zeroenergy. Our starting equation features a position-dependent mass, a matrix potential, and anadditional degree of freedom that can be interpreted either as a magnetic field perpendicularto the plane or a generalized Dirac oscillator interaction. We obtain a number of Darboux-transformed Dirac equations for which the zero energy solutions are exactly known.
Keywords: Dirac equation, Darboux transformation, position-dependent mass, magneticfield, Dirac material
Ever since the experimental realization of graphene [18] there has been a rising interest in Diracmaterials and their applications. The distinguishing feature of Dirac materials such as grapheneis that low-energy charge carriers behave like relativistic massless particles. As such, their dy-namics within a monolayer of the material can be described through the two-dimensional, mass-less Dirac equation. We point out that this is not true anymore if several layers are present, suchas in bilayer graphene [26]. There is a vast amount of literature on Dirac materials and theirapplications, such that we refer the reader to the comprehensive reviews [5] [41] and referencestherein. One of the standing tasks in the field is the confinement of charge carriers within aDirac material, where the effect of Klein tunneling [22] has to be overcome. An overview ofthe problem and resolutions that have been proposed is given in [13] [14]. As pointed out inthe latter references, a variety of methods has been explored for confining relativistic particlesin Dirac materials, including the introduction of a position-dependent mass, and coupling thesystem to magnetic fields. Both of these generalizations have been implemented in Dirac sys-tems. For example, Dirac systems with magnetic fields were studied on a hyperbolic graphenesurface [12], under the presence of nonuniform fields [15], within the minimal-length context[27], among others. Position-dependent masses were used in determining scattering states [7],1ystems with spatially variable Fermi velocity [32] [19] and generalized Dirac oscillators [21].Such oscillators, initially introduced as systems linear in momentum and coordinate variables[29], are closely related to Dirac models coupled to magnetic fields. Applications include exper-imental realizations of Dirac oscillators [17], their coupling to electric fields [23], and within arotating reference frame [39]. Interestingly, it has been shown that in (2 + 1) dimensions theDirac oscillator is equivalent to a spin 1 / m and couplingto a magnetic field and a scalar potential. Effectively, our approach will generate a wide varietyof cases, as the system considered here is equivalent to a generalized Dirac oscillator model or toan inhomogeneous magnetic field. From the application point of view, the m = 0 scenario maybe used to describe motion of electrons in gapless graphene in the presence of electromagneticfields while m = 0 scenario may used for gapped graphene [33] [2]. Let us now briefly discuss themethod we will be using for generating solvable cases of the Dirac model. While the standardDarboux transformation has been extensively applied to the Dirac equation [3] [6] [28] [34] [38],in the present work we apply a different Darboux transformation that was introduced in [40][24] and later reformulated in [36]. This Darboux transformation applies to a specific type ofSchr¨odinger-like equation that can be obtained by suitably decoupling the Dirac equation. Afterapplication of the Darboux transformation we match the resulting Schr¨odinger-type equationto a form that can be put back into Dirac form. The remainder of this work is organized asfollows: section 2 presents a brief review of the Darboux transformation for Schr¨odinger-typeequations we will be using here. In section 3 we construct the generalization of the Darbouxtransformation to our Dirac scenario, while section 4 is devoted to examples. In section 5 weshall consider the same system as in earlier sections except that matrix scalar potentials will beconsidered. Finally, section 6 is devoted to a conclusion. Let us first summarize results from [36]. The starting point is the following pair of Schr¨odinger-type equations ψ ′′ ( x ) − (cid:2) ǫ + ǫ X ( x ) + Y ( x ) (cid:3) ψ ( x ) = 0 (1) ψ ′′ n ( x ) − (cid:2) ǫ + ǫ X n ( x ) + Y n ( x ) (cid:3) ψ n ( x ) = 0 , (2)where the prime denotes differentiation, ǫ is a real-valued constant, the functions X j , Y j , j = 0 , n ,are sufficiently smooth and independent of ǫ , and ψ , ψ n represent the respective solutions fora natural number n . In resemblance to the conventional Schr¨odinger equation we will referto ǫ as energy and to X j , Y j , j = 0 , n , as potential terms. We will now define a Darboux2ransformation that interrelates the two equations (1) and (2). To this end, we define functions v j , j = 0 , ..., n −
1, through v j ( x ) = exp [( ǫ − λ j ) x ] h j ( x ) , j = 0 , ..., n − , (3)where h j , j = 0 , ..., n −
1, are solutions of the initial equation (1) at energies λ j , j = 0 , ..., n − λ , λ ,..., λ n − , ǫ are pairwise different. The solutions h j , j = 0 , ..., n −
1, are called transformation functions. We are now ready to define our n -th orderDarboux transformation. This transformation ψ n of the solution ψ to (1) is given by ψ n ( x ) = W v n − ,ψ ( x ) q ˆ W v n − ( x ) W v n − ( x ) . (4)Here, the quantities W v n − and W n − ,ψ stand for the Wronskians of v , ..., v n − and of v , ..., v n − , ψ , respectively. Furthermore, ˆ W v n − is given byˆ W v n − ( x ) = ( − n G ( x ) W v ,...,v n − ,F ( x ) , where G ( x ) = exp x Z V ( t ) + 2 ǫ dt . (5)The function ψ n solves our transformed Schr¨odinger-type equation (2) if the potential termsmeet the following constraints X n ( x ) = X ( x ) + ddx log " ˆ W v n − ( x ) W v n − ( x ) (6) Y n ( x ) = Y ( x ) − n X ′ ( x ) + X ( x )2 ( ddx log " ˆ W v n − ( x ) W v n − ( x ) + 3 [ ˆ W ′ v n − ( x )] W v n − ( x ) ++ 3 [ W ′ v n − ( x )] W v n − ( x ) − ˆ W ′ v n − ( x ) W ′ v n − ( x )2 ˆ W v n − ( x ) W v n − ( x ) − ˆ W ′′ v n − ( x )2 ˆ W v n − ( x ) − W ′′ v n − ( x )2 W v n − ( x ) . (7)In summary, the quantities (4), (6), and (7) determine the interrelations between the initial (1)and the transformed Schr¨odinger-type equation (2) and their corresponding solutions. The purpose of this section is to construct a Darboux transformation for the two-dimensionalDirac equation at zero energy. The principal idea used for our construction is to connect aninitial and transformed Dirac equation with Schr¨odinger-type counterparts of the form (1) and(2), respectively.
Decoupling the initial Dirac equation.
We start out from the initial Dirac equation in theform { σ x [ p x − i σ z f ( x )] + σ y p y + σ z m ( x ) + V ( x ) I } Ψ( x, y ) = 0 , (8)where σ x , σ y , σ z are the usual Pauli matrices, p x , p y denotes the momentum operators, andΨ is the two-component solution. Furthermore, f can be interpreted as a generalized oscillator3erm, m denotes the position-dependent mass, and V I represents a scalar potential function V multiplied by the 2 × { σ x p x + σ y [ p y − f ( x )] + σ z m ( x ) + V ( x ) I } Ψ( x, y ) = 0 . (9)In this form, our Dirac equation describes a particle that is subjected to a magnetic field [11]:our function f can be interpreted as a component of the vector potential A , given by A ( x ) = (0 , − f ( x ) , T . Consequently, the associated magnetic field B is obtained by applying the curl. This yields B ( x ) = (cid:0) , , − f ′ ( x ) (cid:1) T . (10)For the following it does not make any difference if we consider our Dirac equation in the form(8) or (9), as the only difference between the two forms is the interpretation of the function f .As an example let us mention that the massless case m = 0 of our second form (9) describesa quasiparticle in graphene subjected to an inhomogeneous magnetic field perpendicular tothe graphene sheet. Next, upon inserting the momentum operators and collecting terms, ourequation (8) can be written as follows − i ∂ Ψ( x, y ) ∂x − i ∂ Ψ( x, y ) ∂y + m ( x ) + V ( x ) i f ( x ) − i f ( x ) − m ( x ) + V ( x ) ! Ψ( x, y ) = 0 . (11)Next, noting that the motion in y − direction is free, we introduce the solution components bysetting Ψ( x, y ) = exp( i k y y ) Ψ ( x )Ψ ( x ) ! , (12)where the real-valued constant k y denotes the momentum in the y -direction. Upon implementing(12) in our Dirac equation (11), the spinor components can be shown to follow the followingpair of coupled equations − i Ψ ′ ( x ) + [ − i k y + i f ( x )] Ψ ( x ) + [ m ( x ) + V ( x )] Ψ ( x ) = 0 , (13) − i Ψ ′ ( x ) + [ i k y − i f ( x )] Ψ ( x ) + [ − m ( x ) + V ( x )] Ψ ( x ) = 0 . (14)In order to decouple this system, we solve the second equation with respect to Ψ . This givesΨ ( x ) = [ i f ( x ) − i k y ] Ψ ( x ) + i Ψ ′ ( x ) V ( x ) − m ( x ) . (15)We substitute this setting into the first equation (13), along with the definitionΨ ( x ) = p m ( x ) − V ( x ) ψ ( x ) , (16)introducing a function ψ . This renders our equation (13) in the following form ψ ′′ ( x ) − (cid:2) k y + k y X ( x ) + Y ( x ) (cid:3) ψ ( x ) = 0 , (17)4here the functions X and Y are given by X ( x ) = − f ( x ) − m ′ ( x ) − V ′ ( x ) m ( x ) − V ( x ) (18) Y ( x ) = 14 [ m ( x ) − V ( x )] ( f ( x ) [ m ( x ) − V ( x )] + 4 f ( x ) [ m ( x ) − V ( x )] ×× [ m ′ ( x ) − V ′ ( x )] + 3 [ m ′ ( x ) − V ′ ( x )] + 2 [ m ( x ) − V ( x )] (cid:8) m ( x ) − V ( x )] ×× [ m ( x ) − V ( x ) − f ′ ( x )] − m ′′ ( x ) + V ′′ ( x ) (cid:9)) . (19)We observe that the form of our equation (17) matches its general counterpart (1) if we identifythe parameters ǫ and k y . The transformed Dirac system.
As a consequence of the matching we just completed, ourDarboux transformation becomes applicable to (17). While the transformed solution (4) and itsassociated potential terms (6), (7) can be calculated in a straightforward manner, the remainingtask is to use the latter results in order to set up a transformed Dirac equation of the type (8).More precisely, this transformed Dirac equation reads n σ x h p x − i σ z ˆ f ( x ) i + σ y p y + σ z ˆ m ( x ) + ˆ V ( x ) I o ˆΨ( x, y ) = 0 , (20)where we must determine its transformed solution ˆΨ, the term ˆ f , the position-dependent massˆ m , and the scalar potential function ˆ V , multiplied by the 2 × n σ x p x + σ y h p y − ˆ f ( x ) i + σ z ˆ m ( x ) + ˆ V ( x ) I o ˆΨ( x, y ) = 0 , (21)which we understand to describe a system coupled to a magnetic field that is given byˆ B ( x ) = (cid:16) , , − ˆ f ′ ( x ) (cid:17) T . (22)Next, we will first find the latter three quantities, and afterwards construct the associatedtransformed Dirac solution. After applying the Darboux transformation (4) to equation (17),we obtain a transformed equation of the form (2), that is ψ ′′ n ( x ) − (cid:2) k y + k y X n ( x ) + Y n ( x ) (cid:3) ψ n ( x ) = 0 , (23)recall that k y replaces the parameter ǫ in (2). The potential terms X n and Y n are given by (6)and (7), respectively, where X , Y can be found in (18), (19). Hence, in the case of X n we havethe following explicit form X n ( x ) = − f ( x ) − m ′ ( x ) − V ′ ( x ) m ( x ) − V ( x ) + ddx log " ˆ W v n − ( x ) W v n − ( x ) . (24)It is now important to understand that this expression must be cast in a shape resembling (18),such that it can be linked to our transformed Dirac scenario. This yields the condition − f ( x ) − m ′ ( x ) − V ′ ( x ) m ( x ) − V ( x ) + ddx log " ˆ W v n − ( x ) W v n − ( x ) = − f ( x ) − ˆ m ′ ( x ) − ˆ V ′ ( x )ˆ m ( x ) − ˆ V ( x ) . (25)5he same type of condition must hold for the second potential term Y n . However, since theexplicit form of this condition is very long, as it involves (7) and (19), we omit to state it here.Instead, we give it in abbreviated form as Y n ( x ) = Y ( x ) | f → ˆ f,m → ˆ m,V → ˆ V , (26)The system of equations (25), (26) determines the above mentioned quantities that make up thetransformed Dirac equation (20): the term ˆ f , position-dependent mass ˆ m , and scalar potentialˆ V . We proceed by solving (25) with respect to the transformed term ˆ f . We obtainˆ f ( x ) = 2 f ( x ) + m ′ ( x ) − V ′ ( x ) m ( x ) − V ( x ) − ˆ m ′ ( x ) − ˆ V ′ ( x )ˆ m ( x ) − ˆ V ( x ) − ddx log " ˆ W v n − ( x ) W v n − ( x ) = 2 f ( x ) + ddx log ( [ m ( x ) − V ( x )] W v n − ( x )[ ˆ m ( x ) − ˆ V ( x )] ˆ W v n − ( x ) ) . (27)Note that we could have also solved (25) with respect to the mass or the potential, but thiswould have lead to an unsolvable second condition. For this reason, we go with our function(27). Substitution into the second condition (26) and solving for the transformed scalar potentialgivesˆ V ( x ) = δ (
14 [ m ( x ) − V ( x )] " ˆ m ( x ) + ∆ X ′ n ( x )2 + ∆ X n ( x ) V ′ ( x )2 [ m ( x ) − V ( x )] − m ( x ) + V ( x ) −− − f ( x ) ∆ X n ( x ) + ∆ X n ( x ) − ∆ Y n ( x ) − ∆ X n ( x ) m ′ ( x )2 [ m ( x ) − V ( x )] , (28)where δ = ±
1. For the sake of brevity we used the abbreviations∆ X n ( x ) = X n ( x ) − X ( x ) ∆ Y n ( x ) = Y n ( x ) − Y ( x ) , (29)recall that the quantities involved here are defined in (6) and (7), respectively. Thus, we have nowsolved our system (25), (26) by determining the transformed function (27) and the transformedscalar potential (28). Note that the transformed position-dependent mass remains undeterminedand can be set arbitrarily. It is important to point out that the transformed mass can always bechosen as zero, such that our transformed Dirac equation becomes massless. As mentioned above,this scenario particularly applies to charge carrier transport in Dirac materials like graphene. The transformed Dirac solutions.
It now remains to construct the solution of our trans-formed Dirac equation (20), which we will do in a way similar to its initial counterpart (12). Wedefine the transformed solution in two-component form asˆΨ( x, y ) = exp( i k y y ) ˆΨ ( x )ˆΨ ( x ) ! . (30)The component functions of this solution are interrelated byˆΨ ( x ) = [ i ˆ f ( x ) − i k y ] ˆΨ ( x ) + i ˆΨ ′ ( x )ˆ V ( x ) − ˆ m ( x ) , (31)6ote that this relation is in agreement with (15). It remains to determine the first componentin (30). To this end, let us compare the present case with the initial scenario, where the firstcomponent Ψ of the Dirac solution (12) is linked to a solution ψ of the Schr¨odinger equation(17) by means of (16). This means that in the transformed scenario, the first solution componentˆΨ is related to the transformed Schr¨odinger solution ψ n asˆΨ ( x ) = q ˆ m ( x ) − ˆ V ( x ) ψ n ( x ) , (32)recall that ˆ V is given in (28) and ˆ m is arbitrary. Next, we observe that the function ψ n enteringin (32) is a solution of the transformed Schr¨odinger equation (2). As such, it can be writtenusing the Darboux transformation (4). This renders (32) in the formˆΨ ( x ) = q ˆ m ( x ) − ˆ V ( x ) W v n − ,ψ ( x ) q ˆ W v n − ( x ) W v n − ( x ) . (33)Let us now establish the connection between the functions v j , j = 0 , ..., n − h j , j = 0 , ..., n −
1, of our initial Schr¨odinger equation (17). Upon using the same relation as in(16), we find v j ( x ) = exp [( k y − λ j ) x ] s m ( x ) − V ( x ) χ j ( x ) , j = 0 , ..., n − , (34)where χ j is the first component of a solution to our transformed Dirac equation (20) for k y = λ j , j = 0 , ..., n −
1. The associated second component can be found through the same transformationas used in (15).
We will now present several applications for the Darboux transformation that was constructedin the previous section. While our construction’s starting point is the initial Dirac equation (8),from a practical point of view it is typically more efficient to use our Schr¨odinger-type equation(17) instead. The reason is that solutions of the latter equation can be found much more easilythan of its Dirac counterpart. Once a solution to (17) is known, solutions, potentials, andterms for both our initial and transformed Dirac equation can be generated. We will follow thisprocedure in our subsequent examples. Due to the importance of the initial Schr¨odinger-typeequation (17) for the Darboux transformation we will now mention a particular simplificationthat arises when parameters are chosen suitably. The principal idea of this parameter choice isto remove the term proportional to k y , that is, we impose the condition X = 0 in (18). Thiscondition can be fulfilled by choosing the term as f ( x ) = V ′ ( x ) − m ′ ( x )2 [ m ( x ) − V ( x )] . (35)Upon substituting this into our equation (17), the remaining potential term (19) simplifies. Weobtain ψ ′′ ( x ) + (cid:2) − k y + V ( x ) − m ( x ) (cid:3) ψ ( x ) = 0 . (36)7his equation can be interpreted as a conventional Schr¨odinger equation with energy − k y andpotential m − V . Hence, we can choose the initial mass m and potential V in order to obtaina solvable Schr¨odinger equation (36). The only parameter restriction is that the energy mustbe negative. This is so because the parameter k y must be real-valued due to our definition (12)of the Dirac solution. Let us also point out that the term (35) is determined once the mass m and the potential V have been chosen. Let us consider our initial Dirac equation (8) or, equivalently, the form (9) for the followingparameter settings f ( x ) = 12 tanh( x ) V ( x ) = √
30 sech( x ) m ( x ) = 0 . (37)Note that the factor √
30 in the potential was chosen in order to obtain a certain amount ofbound-state solutions to our Dirac equation, as will be demonstrated below. Observe furtherthat the settings (37) render (8) in massless form, such that it applies to Dirac materials likegraphene. The functions from (37) are shown in the right part of figure 1. While V stands forthe scalar potential, the function f can either denote a generalized oscillator term according to(8) or it can represent a magnetic field within (9) that is found by means of (10) as B ( x ) = (cid:18) , , −
12 sech( x ) (cid:19) T . (38)Hence, the last component of the magnetic field has the shape of a pulse. We substitute oursettings into the Schr¨odinger equation (17) that after simplification takes the form ψ ′′ ( x ) − (cid:2) k y −
30 sech( x ) (cid:3) ψ ( x ) = 0 . (39)We observe here that the term proportional to k y has vanished. This is so because our choiceof parameters in (37) satisfies (35). The general solution of equation (39) can be written as ψ gen ( x ) = c P k y [tanh( x )] + c Q k y [tanh( x )] , (40)where P and Q stand for the associated Legendre function of the first and second kind, respec-tively [1]. In order to simplify calculations and to extract bound-state solutions, we will considerthe following particular solution of equation (39), obtained from the general case (40) by setting c = 1 and c = 0 ψ ( x ) = P k y [tanh( x )] , (41)The function (41) enables us to find a solution to our initial Dirac equation (8) with the settings(37). Upon substitution of (41) into (16) and (15), we obtain the component functions of thesolution (12) as followsΨ ( x ) = p sech( x ) P k y [tanh( x )]Ψ ( x ) = i √ p sech( x ) × (42) × ( ( k y −
6) cosh( x ) P k y [tanh( x )] + [6 sinh( x ) − k y cosh( x )] P k y [tanh( x )] ) . (43)8he corresponding solution (12) of our initial Dirac equation (8) represents bound states if theparameter k y attains integer values in the interval [1 , - - x Y H x , 0 L - - - x V H x L ,f H x L , B z H x L Figure 1: Left plot: graphs of the normalized probability densities | Ψ( x, | associated with thesolution (12) for components (42) and (43). Parameter settings are k y = 5 (black solid curve), k y = 4 (gray curve), and k y = 3 (dashed curve). Right plot: the initial functions V (black solidcurve), f (gray curve) from (37), and the z − component of the magnetic field (38) for the mass m = 0. First-order Darboux transformation.
Let us first perform a transformation of order oneby setting n = 1 throughout (4), (6), and (7). We choose the transformation function h from(41) for the transformation parameter λ = 5, that is, we set h ( x ) = ψ ( x ) | k y =5 = P [tanh( x )] = − (cid:2) − tanh( x ) (cid:3) . (44)We substitute this function into (3) and the Darboux transformation (4), (6), (7), and weafterwards plug the results along with our settings (37) into the transformed scalar potential(28) and the term (27). This gives for the choice δ = − V ( x ) = − p ˆ m ( x ) + 24 sech( x ) ˆ f ( x ) = −
12 + 12 tanh( x ) − ˆ m ′ ( x ) − ˆ V ′ ( x )2 ˆ m ( x ) − V ( x ) , (45)We observe that the these function is defined on the whole real line, provided the mass fuctionˆ m is real-valued and nonnegative. The associated solution of our transformed Dirac equation(20) is obtained through the formulas (31) and (33). We do not state the corresponding generalexpressions in explicit form due to their length. Instead, we give examples for specific massfunctions. In our first example we consider the massless scenario, that is, we setˆ m ( x ) = 0 . (46)This choice renders our scalar potential (28) and the function (27) in the formˆ V ( x ) = − √ x ) ˆ f ( x ) = tanh( x ) − , (47)9here we set δ = −
1. Graphs of these functions are shown in the right part of figure 2. In theform (20) of our Dirac equation, our function ˆ f stands for a generalized oscillator term, whilein the equivalent form (21) we use (22) to determine the magnetic field that is represented byˆ f . We obtain ˆ B ( x ) = (cid:0) , , − sech( x ) (cid:1) T . It remains to construct a solution of our transformed Dirac equation. To this end, we will nowuse (47) to evaluate the components (31) and (33) of our transformed Dirac solution (30). Thisgives usˆΨ ( x ) = 1 √
10 cosh( x ) p [ − − tanh( x )] ×× ( ( k y −
6) cosh( x ) P k y [tanh( x )] + [5 − k y + 11 cosh( x ) + 11 sinh( x )] P k y [tanh( x )] ) (48)ˆΨ ( x ) = − i √
15 cosh( x ) p − − tanh( x ) ×× ( − k y −
6) cosh( x )[(7 − k y ) cosh( x ) P k y [tanh( x )] + [2 ( k y −
2) cosh( x ) −−
19 sinh( x )] P k y [tanh( x )] + { [ − − k y + k y + (72 − k y + k y ) cosh(2 x ) −− k y −
4) sinh(2 x )] } P k y [tanh( x )] ) . (49)Normalized probability densities associated with these solutions are shown in the left part offigure 2. We observe that the solutions are of bound-state type if k y = 1 , , ,
4. In other words,the momentum k y can not take arbitrary values and must necessarily be quantized in order forbound states to exist. Let us now switch to a massive case of our Dirac equation (8) by choosingˆ m ( x ) = sech( x ) . (50)Upon plugging this mass into the transformed scalar potential (28) and our function (27), thelatter quantities are rendered in the formˆ V ( x ) = − x ) ˆ f ( x ) = tanh( x ) − , (51)10 - x Y H x , 0 L - - - x - - - - - V ` H x L , f ` H x L , B ` z H x L Figure 2: Left plot: graphs of the normalized probability densities | ˆΨ( x, | associated with thesolution (30) for components (48) and (49). Parameter settings are δ = − k y = 4 (black solidcurve), k y = 3 (gray curve), and k y = 2 (dashed curve). Right plot: the transformed functions ˆ V (black solid curve), ˆ f (gray curve) from (47), and the z − component of the associated magneticfield (22) (dashed curve) for δ = − δ = −
1. The solution components (31) and (33) evaluate as followsˆΨ ( x ) = s
110 cosh( x ) [ − − tanh( x )] ×× ( ( k y − P k y [tanh( x )] + [5 − k y + 11 tanh( x )] P k y [tanh( x )] ) (52)ˆΨ ( x ) = i − √
10 cosh( x ) p − − tanh( x ) ×× ( k y −
6) cosh( x )[( k y −
7) cosh( x ) P k y [tanh( x )] + [ − k y −
2) cosh( x ) ++ 19 sinh( x )] P k y [tanh( x )] + { ( k y − k y + 6) + [72 + ( k y − k y ] cosh(2 x ) ++ 6 ( − k y + 4) sinh(2 x )] } P k y [tanh( x )] ) . (53)The associated solution (30) represents bound states if k y takes integer values in the interval[1 , h ( x ) = Q . [tanh( x )] . (54)Note that we obtained this transformation function from the general solution (40) by setting c = 0 and c = 1. Upon performing the Darboux transformation (4), (6), (7) for the settings11 - x Y` H x , 0 L - - - x - - - - - V ` H x L , f ` H x L , B ` z H x L Figure 3: Left plot: graphs of the normalized probability densities | ˆΨ( x, | associated with thesolution (30) for components (52) and (53). Parameter settings are δ = − k y = 4 (black solidcurve), k y = 3 (gray curve), and k y = 2 (dashed curve). Right plot: the transformed functions ˆ V (black solid curve), ˆ f (gray curve) from (51), and the z − component of the associated magneticfield (22) (dashed curve) for δ = − m ( x ) = 0 and ˆ m ( x ) = 1 + tanh( x ),we obtain the results shown in figure 4. For the sake of brevity we do not include the explicitexpressions for the quantities shown in figure 4, as they can be obtained in a straightforwardmanner by plugging the chosen mass function into (27) and (28). Generalization and bound states.
We will now generalize the previous example by intro-ducing a nonzero initial position-dependent mass function. Our new settings that replace (37)are given by f ( x ) = 12 tanh( x ) V ( x ) = √
30 sech( x ) m ( x ) = α √
30 sech( x ) . (55)Here, α is a real-valued parameter that controls the strength of the mass function. We observethat the latter function is proportional to the scalar potential. We will comment on this propertybelow in a more general context. The purpose of the present example is to study the effect of α on the transformed system, in particular on the discrete spectrum. To this end, let us substitutethe settings (55) into the Schr¨odinger equation (17). We obtain ψ ′′ ( x ) − (cid:2) k y −
30 sech( x ) (1 − α ) (cid:3) ψ ( x ) = 0 . (56)From the Schr¨odinger perspective, the potential associated with this equation has the form of asingle-well, the depth of which is determined by α . If α vanishes, the well has maximum depth,such that the system supports five bound states [8]. As the value of α increases, the potentialwell’s depth decreases, as well as the number of supported bound states. When α = 1, thepotential vanishes and no bound states are supported by the system. This can be verified bylooking at the actual bound-state solutions of (56). Their general form reads ψ ( x ) = P k y − + √ − α [tanh( x )] , (57)12 - - x - - V ` H x L , f ` H x L , B ` z H x L - - - x - - - - - V ` H x L , f ` H x L , B ` z H x L Figure 4: Graphs of the transformed scalar potential (28) (black solid curves), the term (27)(gray curves), and the z − component of the associated magnetic field (22) (dashed curves) forthe mass functions ˆ m ( x ) = 0 (left plot) and ˆ m ( x ) = 1 + tanh( x ) (right plot) and the settings(37), (54).where the lower index of the Legendre function must be a positive integer, and the upper indexmust be an integer. In addition, (57) must satisfy the condition −
12 + 12 p − α − | k y | = N, N = 0 , , , ..., (58)where | k y | can take integer values in the interval [1 , k y , the numberof solutions to equation (58) decreases as α raises. The values of α that generate a specificnumber of supported bound states is shown in table 1. It is straightforward to verify thatNumber of bound states α q q q q Table 1: Number of bound states and associated parameter value α .the behavior of the bound-state solutions to (56) is the same for our initial Dirac equation.In particular, the values of α given in table 1 remain valid for the initial Dirac case (8). Forthe sake of brevity we omit to show the actual solution. As far as the transformed Diracequation is concerned, the numbers from table 1 are not valid anymore because the number ofsupported bound states depends not only on α , but also on the transformation function usedin the Darboux transformation, and on the transformed position-dependent mass function. The13nly general statement that can be made is that the number of supported bound states decreasesif α increases. Second-order Darboux transformations.
Let us return to our Dirac equation (8) withthe settings (37), and perform a Darboux transformation of second order. This requires twotransformation function h and h that we define as h ( x ) = ψ ( x ) | k y =5 = P [tanh( x )] = − (cid:2) − tanh( x ) (cid:3) h ( x ) = ψ ( x ) | k y =4 = P [tanh( x )] = 945 tanh( x ) (cid:2) − tanh( x ) (cid:3) . (59)Note that h is the same as its counterpart in (44). We now apply our Darboux transformationby substituting (44), (59) into (3) and (4), (6), (7) for n = 2. The results in combination withour settings (37) determine the transformed scalar potential (28) and the function (27). We findˆ V ( x ) = − p ˆ m ( x ) + 18 sech( x ) (60)ˆ f ( x ) = − x ) − ˆ m ′ ( x ) − ˆ V ′ ( x )2 ˆ m ( x ) − V ( x ) , (61)recall that we set δ = −
1. Graphs of these two functions are shown in 5 for specific masses.While the left part of the latter figure displays the massless scenario, in the right part we createa deformation of the graphs around the point x = − - - x - - - - V ` H x L , f ` H x L , B ` z H x L - - x - - - - V ` H x L , f ` H x L , B ` z H x L Figure 5: Graphs of the transformed functions (60) (black solid curve), (61) (gray curve), andthe z − component of the associated magnetic field (22) (dashed curve) for the mass functionsˆ m ( x ) = 0 (left plot) and ˆ m ( x ) = sech( x + 5) (right plot).the function (61) generates a magnetic field that is found by means of (22). For the case ofvanishing mass ˆ m = 0, the latter magnetic field readsˆ B ( x ) = (cid:16) , , − √ x ) (cid:17) T . The solutions of our transformed Dirac equation (20) associated with the quantities (60), (61)are shown in figure 6 as normalized probability densities.14 - x Y` H x , 0 L Figure 6: Graphs of the normalized probability densities | ˆΨ( x, | associated with the solution(30) of our transformed Dirac equation (20) for ˆ m = 0 and the settings (60), (61). Parametervalues are δ = − k y = 3 (black solid curve), k y = 2 (gray curve), and k y = 1 (dashed curve). In this section we will present another example of applying our Darboux transformation to theinitial Dirac equation in any of the equivalent forms (8) or (9). We will choose the followingparameter setting for the initial scenario f ( x ) = 0 V ( x ) = α sech( x ) m ( x ) = 0 , (62)where α is a negative real number. Upon implementation of these settings, our initial Diracequation renders in massless form with the scalar potential V , special cases of which are shownin the right part of figure 4. Now, insertion of the settings (62) into our Schr¨odinger equation(17) renders the latter in the form ψ ′′ ( x ) − (cid:20) k y + k y tanh( x ) + (cid:18) − α (cid:19) sech( x ) + 14 (cid:21) ψ ( x ) = 0 . (63)We observe that in comparison to its counterpart (39), this equation contains a term proportionalto k y because the settings (62) do not comply with the condition (35). Equation (63) is exactly-solvable with particular solution ψ ( x ) = cosh( x ) [1 − tanh( x )] + k y [ − x )] − ky [1 + tanh( x )] + ky ×× F "
12 + k y − q,
12 + k y + q,
32 + k y ,
11 + exp(2 x ) , (64)where F stands for the hypergeometric function [1]. Before we focus on our Darboux transfor-mation, let us construct a solution of our initial Dirac equation (8). To this end, we substitute(64) into the components (16) and (15) of (12). We obtain the resultΨ ( x ) = p sech( x ) ψ ( x ) (65)Ψ ( x ) = − i α p sech( x ) (" k y cosh( x ) + sinh( x ) ψ ( x ) − x ) ψ ′ ( x ) ) , (66)15here the function ψ is defined in (64). The components (65), (66) represent bound states if α and k y are interrelated as 12 + k y + α = − N, N = 0 , , , , ... We observe that this is precisely the condition under which the first argument of the hyper-geometric function in (64) turns into a nonpositive integer. As a result, the latter functiondegenerates to a polynomial. The left part of figure 4 visualizes an example for a specificparameter setting. - - x Y H x , 0 L - - - x - - - - - V H x L Figure 7: Left plot: graphs of the normalized probability densities | Ψ( x, | associated with thesolution (12) for components (65) and (66). Parameter settings are α = − k y = 9 / k y = 7 / k y = 5 / α = − α = − α = − First-order Darboux transformation.
In order to keep calculations simple, we restrictourselves to the case α = − h ( x ) = ψ ( x ) | k y = − = exp (cid:0) x (cid:1) p x ) , (67)where for the sake of simplicity we switched from hyperbolic to exponential functions. In thenext step we plug (67) into (3) and into the Darboux transformation (4), (6), (7) for n = 1.Afterwards we insert the results in combination with our settings (62) into the function (27)and the scalar potential (28). We obtainˆ V ( x ) = − s ˆ m ( x ) + 12 exp(2 x )[3 + exp(2 x )] (68)ˆ f ( x ) = −
12 + 33 + exp(2 x ) − ˆ m ′ ( x ) − ˆ V ′ ( x )2 ˆ m ( x ) − V ( x ) . (69)As in the previous occurrences we have set δ = −
1. If the mass ˆ m is regular on the whole realline, so are the two functions (68) and (69) because the denominators are nonnegative. Figure16 shows graphs of the transformed quantities ˆ V and ˆ f for two particular mass choices. Weobserve that the first of these choices ˆ m = 0 makes the term (69) vanish. We omit to show - - - x - - - - - V ` H x L , f ` H x L - - - x - - V ` H x L , f ` H x L , B ` z H x L Figure 8: Graphs of the transformed scalar potential (68) (black solid curves), the transformedgeneralized oscillator term (69) (gray curve), and the z − component of the associated magneticfield (22) (dashed curve) for the mass functions ˆ m ( x ) = 0 (left plot) and ˆ m ( x ) = exp( − x / δ = − Second-order Darboux transformation.
Let us now apply a Darboux transformation ofsecond order to our initial Dirac equation (8) for the parameter settings (62). We need twotransformation functions h and h that we define as follows h ( x ) = ψ ( x ) | k y = − = exp (cid:0) x (cid:1) p x ) h ( x ) = ψ ( x ) | k y = − = exp (cid:0) x (cid:1) p x ) , (70)observe that we took h from (67). Now, we insert our two transformation functions into (3)and calculate the Darboux transformation (4), (6), (7) for n = 2. The resulting expressions,along with the present parameter settings (62) are then substituted into the term (27) and thescalar potential (28). Simplification and setting δ = − V ( x ) = − s ˆ m ( x ) + 20 exp(2 x )[5 + exp(2 x )] (71)ˆ f ( x ) = −
12 + 55 + exp(2 x ) − ˆ m ′ ( x ) − ˆ V ′ ( x )2 ˆ m ( x ) − V ( x ) . (72)Comparison of these expressions with their first-order counterparts (68) and (69) shows that theydiffer merely in constants. This is due to the choice of our transformation energies as negativeintegers that render the transformation functions in elementary form. We omit to show graphsof the functions (71) and (72) because they are so similar to (68) and (69), respectively. Also, we17o not display the explicit form of solutions pertaining to the transformed Dirac equation (20)for (71) and (72). Instead, we repeat our second-order Darboux transformation with complexconjugate transformation energies. More precisely, we choose our transformation functions as h ( x ) = ψ ( x ) | k y = − i = exp (cid:2)(cid:0) − i (cid:1)(cid:3) x p x ) (73) h ( x ) = ψ ( x ) | k y = − − i = exp (cid:2)(cid:0) + i (cid:1)(cid:3) x p x ) . (74)Following our previous procedure, we substitute these two functions into (3), and afterwardsinto the Darboux transformation (4), (6), (7) for n = 2, which in turn determines the term (27)and the scalar potential (28). We find for δ = − V ( x ) = − s ˆ m ( x ) + 260 exp(2 x )[13 + 5 exp(2 x )] (75)ˆ f ( x ) = −
12 + 1313 + 5 exp(2 x ) − ˆ m ′ ( x ) − ˆ V ′ ( x )2 ˆ m ( x ) − V ( x ) . (76)The form of these functions is the same as the previous pairs (71), (72) and (68), (69). Examplesare shown in figure 9 for two different masses. Note that the first mass choice ˆ m = 0 makes theterm (76) vanish. - - x - - - - - V ` H x L , f ` H x L - x - - V ` H x L , f ` H x L , B ` z H x L Figure 9: Graphs of the transformed scalar potential (75) (black solid curves), the transformedgeneralized oscillator term (76) (gray curve), and the associated z − component of the magneticfield (22) (dashed curve) for the mass functions ˆ m ( x ) = exp( − x ) (left plot) and ˆ m ( x ) = 0 (rightplot) and the settings (62), (73), (74), δ = − In this section we shall apply Darboux transformation to a more general relativistic system,namely, Dirac equation in the presence of a matrix potential [9] [16] [37] [35] and find new18atrix potentials for which the Dirac equation remains solvable. More precisely, we considerour initial Dirac equation in the form { σ x [ p x − i σ z f ( x )] + σ y p y + σ z m ( x ) + V ( x ) } Ψ( x, y ) = 0 , (77)where we use the same notation as in (8), except that this time the potential V = ( V ij ), i, j = 1 , × − i ∂ Ψ( x, y ) ∂x − i ∂ Ψ( x, y ) ∂y + m ( x ) + V ( x ) i f ( x ) + V ( x ) − i f ( x ) + V ( x ) − m ( x ) + V ( x ) ! Ψ( x, y ) = 0 . (78)In the forms (77) and (78), the function f can be interpreted as a generalized oscillator term anda component of a vector potential, respectively. In the latter case the associated magnetic fieldis found from (10). We will now approach our initial equation (77) or, equivalently, its form (78)in the same way as their respective counterparts (8) and (9) in section 3. In each step we canrecover the latter particular case if we implement the settings V = V = V , V = V = 0.Let us now substitute (12) into (78), resulting in the component equations − i Ψ ′ ( x ) + [ − i k y + i f ( x ) + V ( x )] Ψ ( x ) + [ m ( x ) + V ( x )] Ψ ( x ) = 0 (79) − i Ψ ′ ( x ) + [ i k y − i f ( x ) + V ( x )] Ψ ( x ) + [ − m ( x ) + V ( x )] Ψ ( x ) = 0 . (80)We solve the second component equation with respect to Ψ . This yieldsΨ ( x ) = [ i f ( x ) − i k y − V ( x )] Ψ ( x ) + i Ψ ′ ( x ) V ( x ) − m ( x ) . (81)The remaining component (79) can be rewritten by redefining Ψ asΨ ( x ) = exp − i x Z V ( t ) + V ( t ) dt p m ( x ) − V ( x ) ψ ( x ) . Upon implementing this definition in (79), we obtain the following Schr¨odinger-type equationfor the function ψ ψ ′′ ( x ) − (cid:2) k y + k y X ( x ) + Y ( x ) (cid:3) ψ ( x ) = 0 , (82)where the potential term X is given explicitly by X ( x ) = − f ( x ) + i [ V ( x ) − V ( x )] − m ′ ( x ) − V ′ ( x ) m ( x ) − V ( x ) . (83)Since the remaining potential term Y has a very long and involved form, we omit to state itexplicitly here. Before we continue, let us briefly comment on a simplification of our Schr¨odinger-type equation (82) that occurs for X = 0. Similar to the setting (35) worked out in the previoussection, we fix our term f to be given as f ( x ) = − m ′ ( x ) − V ′ ( x )2 m ( x ) − V ( x ) + i V ( x ) − V ( x )] . (84)This setting forces X = 0 and furthermore renders our equation (82) in the compact form ψ ′′ ( x ) + (cid:8) − k y + [ m ( x ) + V ( x )] [ m ( x ) − V ( x )] (cid:9) ψ ( x ) = 0 . (85)19e observe that this generalization of (36) resembles a conventional Schr¨odinger equation, where − k y plays the role of the stationary energy. Now let us return to our Darboux transformation.After applying the latter transformation (4), (6), (7) to equation (82), we obtain its transformedcounterpart as ψ ′′ n ( x ) − (cid:2) k y + k y X n ( x ) + Y n ( x ) (cid:3) ψ n ( x ) = 0 . (86)Our next step consists in matching the form of the transformed potential terms with their initialpartners. Our goal is to transfer (86) to our transformed Dirac equation n σ x h p x − i σ z ˆ f ( x ) i + σ y p y + σ z ˆ m ( x ) + ˆ V ( x ) o ˆΨ( x, y ) = 0 , (87)where we adopt the notation from (20) except for the transformed potential ˆ V = ( ˆ V ij ), i, j = 1 , X n reads − f ( x ) + i [ V ( x ) − V ( x )] − m ′ ( x ) − V ′ ( x ) m ( x ) − V ( x ) + ∆ X n ( x ) == − f ( x ) + i [ ˆ V ( x ) − ˆ V ( x )] − ˆ m ′ ( x ) − ˆ V ′ ( x )ˆ m ( x ) − ˆ V ( x ) . (88)Furthermore, note that we implemented the abbreviation ∆ X n from (29). We can solve ourcondition (88) with respect to the term ˆ f asˆ f ( x ) = f ( x ) + i h ˆ V ( x ) − ˆ V ( x ) i − i V ( x ) − V ( x )] + m ′ ( x ) − V ′ ( x )2 m ( x ) − V ( x ) ++ h ˆ V ( x ) − ˆ m ( x ) i ∆ X n ( x ) − ˆ m ′ ( x ) + ˆ V ′ ( x )2 ˆ m ( x ) − V ( x ) . (89)Next we must solve the remaining condition pertaining to the potential term Y n in (86). Sincewe avoid to state Y n explicitly, we give the latter condition in abbreviated form as Y n ( x ) = Y ( x ) | f → ˆ f,m → ˆ m,V ij → ˆ V ij . (90)We point out that the function ˆ f is given by (89). Upon insertion of this function we can solvecondition (90) with respect to ˆ m , ˆ V , and ˆ V . We cannot use the off-diagonal potential matrixentries ˆ V or ˆ V to solve (90) because they do not occur in our condition. Let us now state the20hree solutions mentioned above. When solving for the mass function ˆ m , we obtainˆ m ( x ) = ˆ V ( x ) V ( x ) − V ( x ) ˆ V ( x ) − ˆ V ( x ) m ( x ) + ˆ V ( x ) m ( x )2 m ( x ) − V ( x ) ++ 12 m ( x ) − V ( x ) ( [ m ( x ) − V ( x )] ( m ( x ) + 4 m ( x ) " V ( x ) − V ( x ) −− ˆ V ( x ) V ( x ) + 4 V ( x ) V ( x ) − V ( x ) ˆ V ( x ) − V ( x ) ˆ V ( x ) −− f ( x ) V ( x ) ∆ X n ( x ) + 2 i V ( x ) V ( x ) ∆ X n ( x ) − i V ( x ) V ( x ) ∆ X n ( x ) ++ V ( x ) ∆ X n ( x ) − V ( x ) ∆ Y n ( x ) + 2 ∆ X n ( x ) m ′ ( x ) − X n ( x ) V ′ ( x ) ++ m ( x ) " − V ( x ) V ( x ) + 4 V ( x ) + ˆ V ( x ) + ˆ V ( x ) ! + ∆ X n ( x ) ×× f ( x ) − i V ( x ) − ∆ X n ( x ) ! + 4 ∆ Y n ( x ) − X ′ n ( x ) ++ 2 V ( x ) ∆ X ′ n ( x ) )) . Let us now solve our condition (90) with respect to the transformed matrix potential entry ˆ V .Our result readsˆ V ( x ) = 14 [ m ( x ) − V ( x )] [ ˆ m ( x ) − ˆ V ( x )] ( m ( x ) + 4 m ( x ) V ( x ) −− m ( x ) V ( x ) + 4 ˆ m ( x ) V ( x ) + 4 V ( x ) V ( x ) −− m ( x ) V ( x ) ˆ V ( x ) − f ( x ) V ( x ) ∆ X n ( x ) ++ 2 i V ( x ) V ( x ) ∆ X n ( x ) − i V ( x ) V ( x ) ∆ X n ( x ) + V ( x ) ∆ X n ( x ) −− V ( x ) ∆ Y n ( x ) + 2 ∆ X n ( x ) m ′ ( x ) − X n ( x ) V ′ ( x ) + 2 V ( x ) ∆ X ′ n ( x ) −− m ( x ) V ( x ) V ( x ) + 4 m ( x ) V ( x ) − m ( x ) ˆ m ( x ) ++ 4 m ( x ) ˆ m ( x ) ˆ V ( x ) + 4 m ( x ) f ( x ) ∆ X n ( x ) −− i m ( x ) ∆ X n ( x ) [ V ( x ) − V ( x )] − m ( x ) ∆ X n ( x ) + 4 m ( x ) ∆ Y n ( x ) −− m ( x ) ∆ X ′ n ( x ) ) (91)As mentioned above, we can also solve condition (90) for the transformed matrix potential entryˆ V . However, the solution is very similar to (91) in the following sense: if we replace ˆ V in(91) by − ˆ V , then we obtain the solution of (90) with respect to ˆ V . For this reason we willnot state its explicit form here. First-order Darboux transformation.
In this paragraph we will demonstrate how ourDarboux transformation works in practice if the potential in our Dirac equation (77) is not a21ultiple of the identity matrix. To this end, let us first specify our initial parameter settings. f ( x ) = 12 tanh( x ) V ( x ) = √
30 sech( x ) I m ( x ) = 0 . (92)We observe that these settings are the same as (37), note that our notation has changed dueto V now being an actual matrix. Consequently, our initial Dirac equation (77) for the settings(92) is the same as its former counterpart (8) with (37). We can therefore use the Schr¨odingersolution (41) and the transformation function (44) for our Darboux transformation. We firstsubstitute the latter two function along with (3) into (4), (6), (7) for n = 1. In the subsequentstep we insert the results into the transformed term (89) . Simplification leads to the findingsˆ f ( x ) = 12 ˆ m ( x ) − V ( x ) ( ˆ m ( x ) " − x ) + i ˆ V ( x ) − i ˆ V ( x ) + ˆ V ( x ) −− " tanh( x ) ˆ V ( x ) + i ˆ V ( x ) ˆ V ( x ) − i ˆ V ( x ) ˆ V ( x ) − ˆ m ′ ( x ) + ˆ V ′ ( x ) ) . (93)In a similar way we can determine the transformed potential matrix ˆ V by substitution of ourcurrent parameters into (91). We obtainˆ V ( x ) = − ˆ m ( x ) + 24 sech( x ) ˆ V ( x ) − ˆ m ( x ) ˆ V ( x )ˆ V ( x ) ˆ V ( x ) . (94)We observe that the transformed mass function and three entries of the transformed potentialmatrix remain undetermined, allowing to generate a wide variety of Dirac equations (87), alongwith its associated solutions. Let us now state an example by introducing the settingsˆ m ( x ) = 1 + tanh( x ) ˆ V ( x ) = − x ) ˆ V ( x ) = ˆ V ( x ) = 0 . (95)If we plug these settings into the term (93), we obtain its explicit formˆ f ( x ) = − sech( x ) [2 + sech( x ) − x )]1 + 4 sech( x ) + tanh( x ) . (96)The magnetic field (22) generated by this function can be calculated asˆ B ( x ) = (cid:18) , , x )[4 + exp( x )] − − x ) + exp( x )] (cid:19) T . (97)The z − component of the magnetic field is visualized in the right part of figure 10. The trans-formed matrix potential is found by inserting our current settings (92) and (95) into (94). Theresulting potential has the formˆ V ( x ) = − [25 + 4 sech( x ) −
22 tanh( x )] [1 + tanh( x )]1 + 4 sech( x ) + tanh( x ) 00 − x ) . (98)Both the term (96) and the non-vanishing potential components from (98) are shown in the rightpart of figure 10. Since the explicit form of the associated solutions to the transformed Diracequation (87) is very long, we omit to show it here. Instead, we visualize the correspondingprobability densities in the left part of figure 10.22 - x Y` H x , 0 L - - - x - - - - - V ` H x L , V ` H x L , f ` H x L , B ` z H x L Figure 10: Left plot: graphs of the normalized probability densities | ˆΨ( x, | associated with thesolution (30) of our transformed Dirac equation (87) for the settings (92), (41), (44). Parametervalues are k y = 4 (black solid curve), k y = 3 (gray curve), and k y = 2 (dashed curve). Rightplot: graphs of the entries ˆ V (black solid curve), ˆ V (gray curve) pertaining to the matrix(98), the oscillator term (96) (black dashed curve), and the z − component of the magnetic field(97) (gray dashed curve). The Darboux transformation presented in this work is applicable to Dirac equations at zeroenergy with magnetic field, position-dependent mass and matrix potential, including the specialcases of vanishing mass and scalar potential. Instead of being coupled to a magnetic field, oursystems can also be interpreted as generalized Dirac oscillators due to a one-to-one correspon-dence between the two scenarios. A particular feature of our approach is that the position-dependent mass in the Darboux-transformed Dirac equation remains undetermined and can bechosen arbitrarily. This property is useful for example when comparing exactly-solvable mass-less systems (such as in Dirac materials) to their massive counterparts. It should be pointedout that the algorithm summarized in section 2 is not equivalent to the conventional Darbouxtransformation,also referred to as SUSY formalism. As such, the results we obtain here cannotbe found through application of the latter formalism. The extension of the present method tomore general systems like bilayer graphene is subject of future research.
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