Exact solution of the 1D Dirac equation for a pseudoscalar interaction potential with the inverse-square-root variation law

We present the exact solution of the one-dimensional stationary Dirac equation for the pseudoscalar interaction potential, which consists of a constant and a term that varies in accordance with the inverse-square-root law. The general solution of the problem is written in terms of irreducible linear combinations of two Kummer confluent hypergeometric functions and two Hermite functions with non-integer indices. Depending on the value of the indicated constant, the effective potential for the Schrödinger-type equation to which the problem is reduced can form a barrier or well. This well can support an infinite number of bound states. We derive the exact equation for the energy spectrum and construct a rather accurate approximation for the energies of bound states. The Maslov index involved turns out to be non-trivial; it depends on the parameters of the potential.


Potential
We consider a spin-1/2 Dirac fermion of rest mass m and energy E in the field of a pseudoscalar interaction potential given as where W 0 and W 1 are arbitrary constants and x is a space coordinate.The stationary one-dimensional Dirac equation for such an interaction can be written as where ψ = (ψ 1 , ψ 2 ) is the two-component wavefunction of the particle, σ 0 is the identity matrix, σ 1,2,3 are the Pauli matrices: c is the speed of light, and is the reduced Planck constant.In explicit form, the Dirac equation is written as Eliminating one of the components from this system, one can obtain a one-dimensional Schrödinger-like equation.For instance, resolving the second equation with respect to ψ 2 : where the prime denotes differentiation, we obtain the following Schrödinger-type equation: where we have denoted and introduced the effective Schrödinger potential Potential (1) is a version of the field configuration that was not discussed in previous studies of potentials with the same functional form, including terms that vary according to the law of inverse square root (see 19,20 ).At first glance, the modification appears insignificant, with the only difference being the addition of a constant, W 0 .However, this constant has a dramatic effect on the behavior of the system.Without W 0 , the potential does not support bound states.However, for a negative value of W 0 W 1 , the potential supports infinitely many bound states, regardless of the values of the parameters W 0 and W 1 .This is because in this case the potential becomes a long-range well that can trap particles from a long distance away.Importantly, for a definite-parity extension of potential (1) to the region x < 0 , these observations hold for the entire x-axis.
For an odd-parity extension (Fig. 1), we have and the effective Schrödinger potential is given as The form of this potential is shown in Fig. 2. As we can see, it is symmetric with respect to the origin.As the origin is approached, the potential diverges as |x| −3/2 .And as x goes to infinity, the potential behaves as Hence, it is a long-range potential.For W 1 W 0 > 0 , the potential forms a barrier (Fig. 2, left panel), and for W 1 W 0 < 0 , the potential forms a well (Fig. 2, right panel).In the latter case, the potential supports infinitely many bound states.
We conclude this section by noting that the ability of the background constant pseudoscalar potential to significantly impact the behavior of the system has been observed in several cases (see, for instance 21,22 ).

General solution
The effective Schrödinger potential (11) is a particular case of the first Exton potential (see Eq. ( 21) of 23 ): Here, V 0,1,3 are arbitrary constants.This potential, in turn, is a specific case of one of the five biconfluent Heun potentials originally discussed by Lemieux and Bose 24 (also see 25 ): In general, the solution of the Schrödinger equation for this Lemieux-Bose potential is written in terms of the biconfluent Heun functions, which are advanced special functions [26][27][28] that generalize the Kummer confluent hypergeometric function.This function is encountered in various branches of contemporary research, ranging from classical physics and quantum mechanics to general relativity and cosmology (see [29][30][31][32][33][34] and references therein).Although the biconfluent Heun function is generally a complicated mathematical object, it can be simplified and expressed in terms of simpler mathematical functions, particularly confluent hypergeometric  functions, for certain specifications of its potential.Many of these cases are obtained by terminating the expansion of the biconfluent Heun function in terms of the Hermite functions 35,36 .The first Exton potential ( 13) represents one such case.In this case, the solution is obtained by terminating the expansion at the second term.This simplified solution has been discussed in several papers [37][38][39] .For x > 0 , the effective Schrödinger potential given by Eq. ( 11) represents the first Exton potential with the specification With this, the general solution of the Dirac equation can be written as 39 and C 1 ,C 2 are arbitrary constants.We note that both signs for δ are applicable; however, for definiteness, below we apply the plus sign.From Eqs. ( 16) and ( 17), we observe that each of the two independent fundamental solutions is given as an irreducible linear combination with non-constant coefficients of two Kummer hypergeometric functions or two Hermite functions with non-integer indices.
It can be verified that the wave function ψ 1 vanishes as x goes to infinity only if C 2 = 0 .With this, the solu- tion for x > 0 is simplified to To complete this section, we recall that the second component of the wave function is

Bound states
The possibility of bound states is determined by the boundary conditions at x = 0 .Let the solution for x < 0 that vanishes at infinity be ψ L = (ψ 1L , ψ 2L ) and it be proportional to a constant, say C 3 .Then the continuity condition of the wave function: presents a set of two homogeneous linear equations with respect to the constants C 1 and C 3 .For a non-trivial solution, the determinant of this system must be zero: Since we consider the odd extension, when W(−x) = −W(x) , the Dirac equation is covariant under the parity transformation x → −x , and the solution for the negative x-region can be written as ) .As a result, Eq. ( 24) reduces to ψ 1R (0)ψ 2R (0) = 0 , and we obtain two branches of bound states, generated by ψ 1R (0) = 0 or ψ 2R (0) = 0 .Let us consider these cases separately.
Bound states with ψ 1R (0) = 0.The equation for the bound states' energy spectrum is The behavior of F as a function of energy is shown in Fig. 3.It can be shown that, provided W 0 W 1 < 0 , the spectrum equation has infinitely many roots, all located in the intervals (15) Vol www.nature.com/scientificreports/ To construct an approximation for the spectrum, it is convenient to transform Eq. ( 25), using the recurrence relations between contiguous Hermite functions 40 , into the form The advantage of this form is that the argument y 0 of the involved Hermite functions here is such that it belongs to the left transient region y 0 ≈ √ 2ν − 1 , where ν = a + 1 or ν = a + 2 .Then, using the Airy-function approximation of the Hermite function for this region 41 , we arrive at an approximation of this equation as where It is shown that A 0,1 < 1 and for the solution of Eq. ( 29), a > 1 .Furthermore, for large a , the energy E is close to E 0 : E ≈ E 0 .With these observations, for large a, we arrive at the approximate equation The dependence of a on E (see Eq. ( 19)) shows that this is a sextic polynomial equation in E that is readily reduced to a cubic one.The real roots of the equation are given as where, using Cardano's formula for the cubic, (27)   www.nature.com/scientificreports/This is a fairly accurate approximation.It provides the spectrum with relative error of the order of 10 −4 or less (see Table 1 for a comparison with the exact numerical result).The normalized wave function on the entire x-axis is shown in Fig. 4 ( n = 3 , positive energy branch).It can be observed that the wave function is anti-symmetric with respect to the origin, as expected, and that the derivative of ψ 2 (x) is discontinuous at the origin.
Bound states with ψ 2R (0) = 0.This time, the exact spectrum equation is written as Acting now essentially in the same way as in the previous case, we arrive at the spectrum expressed by the same formulas ( 34)- (36), with the parameter A 1 given as where u is given by Eq. ( 32).The obtained result again is a fairly good approximation as seen from Table 2.The normalized wave function on the entire x-axis is shown in Fig. 5 ( n = 3 , positive energy branch).It can be observed that this time the derivative ψ 1 (x) is discontinuous at the origin.
As Eq. ( 36) shows, the Maslov index is given as where the parameter A 1 is different for the spectrum branches with ψ 1 (0) = 0 and ψ 2 (0) = 0 .An interesting observation is that A 1 is not a constant but depends on the potential parameters W 0 and W 1 .Figure 6 shows this dependence.As we can see, the Maslov index for the energy spectrum branch with ψ 1 (0) = 0 starts from −1/6 , while that for the branch with ψ 2 (0) = 0 starts from −5/6 as u = 0 .We note that both indices tend to −1 as u → ∞. ( Table 1.Comparison of approximation (34) with the numerical solution of exact Eq. ( 27).

Discussion
Thus, we have presented the exact solution of the one-dimensional stationary Dirac equation for a pseudoscalar potential consisting of an independently variable constant and a term that varies in accordance with the inverse square root law.Since the strength of the term that varies in accordance with the inverse square root law can also be independently varied, this is an exactly solvable potential.We have expressed the general solution of the problem as a linear combination with arbitrary coefficients of two fundamental solutions.Each of these fundamental solutions can be expressed as an irreducible linear combination of two functions of the hypergeometric class, namely Kummer hypergeometric functions or Hermite functions with non-integer indices.
A peculiarity of the potential we discussed is that the effective potential for the Schrödinger-like equation, to which the problem is reduced, changes its nature depending on the value of the involved constant term.With   1 /(−W 0 c ) 2/3 .The solid line shows the index for the energy spectrum branch with ψ 1 (0) = 0 and the dotted line stands for the branch with ψ 2 (0) = 0 .The points in the vertical axis indicate −1/6 and −5/6.
certain values, the effective potential is a barrier, and with others, it becomes a well, supporting infinitely many bound states.
We have derived the exact equation for the energy spectrum and have shown that the discrete energies of the bound states are located in two energy intervals separated by the gap (−mc 2 , mc 2 ) .We have constructed a rather accurate approximation for the energies of bound states and calculated the Maslov index of the spectrum.It turns out that this index, which is a constant addition to the quantum number n numbering the bound states and does not vanish at n → ∞ , is rather nontrivial.In the Schrödinger case with a potential, which varies according to the inverse-square-root law, this index is known to be a constant equal to −1/6 17,18 .On the contrary, in our case, the Maslov index turns out to be dependent on the potential parameters and may vary over a rather large interval.
The interaction we have discussed can serve as a model for studying relativistic quantum systems in one dimension, providing insight into confinement effects.The non-polynomial Hermite functions involved in the general solution, confinement behavior, and localization of wave functions near the origin contribute to a comprehensive understanding of relativistic quantum systems and hold promise for diverse applications in condensed matter physics and quantum technology.For example, this interaction can be used to model graphene nanoribbons 42,43 , Weyl and Dirac semimetals 44 , topological insulators and superconductors 45,46 , or quantum dots 47 .

Figure 6 .
Figure 6.The dependence of the Maslov index on the parameter u = W 4/3