## Abstract

Relativistic and non-relativistic scattering by short-range potentials is investigated for selected problems. Scattering by the *δ*′ potential in the Schrödinger equation and *δ* potentials in the Dirac equation must be solved by regularization, efficiently carried out by a perturbation technique involving a stretched variable. Asymmetric regularizations yield non-unique scattering coefficients. Resonant penetration through the potentials is found. Approximative Schrödinger equations in the non-relativistic limit are discussed in detail.

## 1. Introduction

The mathematical base of soliton theory is the inverse-scattering method [1], which is a scheme for constructing general solutions to nonlinear integrable evolution equations through a series of linear computations.

Consider the scattering problem for the time-independent Schrödinger equation with a potential. From textbooks on quantum mechanics, it is well known that this model of tunnelling leads to a spectrum with a discrete part and a continuous part. By the inverse-scattering method, the corresponding bound and free states are mapped into solitons and radiation, respectively. Choosing a Dirac *δ* function as a short-range potential in the Schrödinger equation, we may thus obtain a single soliton and radiation solution to the Korteweg–de Vries equation.

Having worked with nonlinear scattering problems for two decades, we asked the question: what will happen if the *δ* function in the potential is replaced with its derivative? This is not a textbook example. Unlike the *δ* scattering problem, this problem cannot be solved by integration. Instead regularization of the *δ*′ potential must be applied.

In the study of Christiansen *et al*. [2], we consider this problem and find that the potential is penetrable when a resonance condition for the square root of the amplitude of the *δ*′ function is satisfied. However, the solution obtained by regularization is not unique and has been studied in more detail by Zolotaryuk *et al*. [3] and Zolotaryuk [4]. Scattering potentials with a higher degree of singularity than the *δ*′ potential were considered by Exner & Kondej [5], Zolotaryuk *et al*. [6] and Zolotaryuk [7].

In recent decades, many mathematicians have treated similar point interaction problems; e.g. Albeverio *et al*. [8] and references therein, as well as in the study of Christiansen *et al*. [2].

In the present paper, we consider the relativistic case. Here, the Schrödinger equation is replaced by the Dirac equation. In one spatial dimension, the Dirac four-vector reduces to a two-vector. The two field components then satisfy a set of two coupled first-order differential equations (e.g. [9]). Scattering problems for the *δ* potential in the Dirac equation have been solved in the studies of Calkin *et al*. [10] and Benguria *et al*. [11] in a number of interesting cases.

This paper is structured as follows.

In §2, we revisit the *δ* and *δ*′ Schrödinger equation scattering problems, using in the latter case a perturbation techniques suggested to us by O. Penrose (2005, private communication). By introducing a stretched coordinate, one efficiently solves the *δ*′ problem for the second-order differential equation. The trick works when the field derivative is one order higher than the derivative of the *δ* function in the potential.

In §3, we introduce the Dirac equation and choose the Dirac matrices among any two of the Pauli matrices, and the single component of the potential, including the identity matrix or one of the Pauli matrices, such that a set of uncoupled Schrödinger equations are obtained by differentiation. A systematic treatment of the non-relativistic limit by the Foldy–Wouthuysen transformation (e.g. [12]) is not included in our paper.

Section 4 presents solutions to the two resulting types of Dirac equations, obtained by O. Penrose’s (2005, private communication) techniques.

The non-relativistic limit, where the velocity of light , is considered in §5. The two types of solutions to the Dirac equation are compared with solutions to the corresponding Schrödinger equations, which we find—in the case of *δ* potentials—can only be approximative.

Finally, in §6, our results are listed and discussed.

## 2. The *δ* and the *δ*′ scattering problem using Penrose’s techniques

We first recall the text-book example: solution of the time-independent Schrödinger equation, normalized with . In one spatial dimension, the wave function, *ψ*(*x*), satisfies
2.1
where *k*^{2} denotes the energy, with *k* being the propagation constant, and the potential *U*(*x*) is given by
2.2
with *δ*(*x*) being Dirac’s *δ* function and *ρ* denoting the amplitude of the potential. Integrating across the singularity at *x*=0, the conditions
2.3
where the prime denotes differentiation with respect to *x*, are immediately obtained.

Writing the field as the sum of an incident and a reflected field for *x*<0 and a transmitted field for *x*>0,
2.4
the conditions (2.3) yield the reflection and transmission coefficients
2.5a
and
2.5b
Apart from the continuous spectrum, there is a discrete spectrum consisting of a simple pole at *k*=−i*ρ*/2.

In soliton theory, the inverse-scattering method [1] maps the discrete and continuous parts into solitons (a single soliton, in the present case) and radiation, respectively.

In the study of Christiansen *et al*. [2], we investigated the scattering problem, where the potential is given by the derivative of the *δ*(*x*) function,
2.6
where the amplitude of the potential is denoted by *τ*^{2} for convenience.

When an electromagnetic wave propagates in a medium with a two-layered structure of the form
2.7
where *n*(*x*) is the refractive index and *n*,*d* and *θ* (*d*>*θ*) are constants, the wave equation may be reduced to a Schrödinger equation with a *δ*′(*x*) potential, as we shall see from the regularization of this potential given by equations (2.8) and (2.9). This situation is realized from the study of Hooper *et al*. [13], in which the possible transparency of (metallic) tunnel barriers is demonstrated experimentally.

Integration of equation (2.1) with the potential (2.6) immediately leads to a contradiction. Therefore, the problem must be solved by regularization. For this purpose, we shall here use a perturbation technique suggested to us by O. Penrose (2005, private communication).

Introducing the stretched coordinate *ξ*=*x*/*ϵ*, we regularize the potential (2.6) by
2.8
where *ϵ* is a small perturbation parameter, *ϵ*>0, and the function *u*(*ξ*), as shown in figure 1, is given by
2.9
where the parameters *μ*>0 and *η*>0. **Δ**′(*x*), given by equations (2.8) and (2.9), is clearly a regularization of *δ*′(*ξ*), since the areas of the rectangles above and below the *x*-axis are constantly equal to unity under the limiting processes, and . For *μ*≠*η*, *u*(*ξ*) becomes an asymmetric function. We open this possibility in order to demonstrate the non-uniqueness of the regularization procedure.

Inserting equation (2.8) into (2.1), we obtain 2.10 in the limit .

The solution to equation (2.10) may be written in the form
2.11
where *A*_{1},*A*_{2},*B*_{1} and *B*_{2} are integration constants.

For *x*<−*μϵ* and *x*>*ηϵ*, the incident and reflected fields and the transmitted field, respectively, may still be written in the form of equation (2.4).

Requiring continuity of *ψ* and *ψ*′ at *x*=−*ϵμ*, 0 and *x*=*ϵη* yields six conditions for six unknown coefficients *R*,*T*,*A*_{1},*A*_{2},*B*_{1} and *B*_{2}. As a result, we find
2.12
for , and
2.13a
and
2.13b
for .

Thus, the values of *R* and *T* depend on the choice of the regularization parameters *μ* and *η*.

In the anti-symmetric case, *μ*=*η*, we still get equation (2.12) for . Without loss of generality, we may let *μ*=1, finding for
2.14a
and
2.14b
as in the study of Christiansen *et al*. [2].

Thus, there exists a countable discrete set of values of *τ*, i.e. the square root of the amplitude of the *δ*′ function, fulfilling the resonance condition . For these values of *τ*, partial transmission through the *δ*′ potential occurs. Note that the reflection and transmission coefficients, *R* and *T*, given by equation (2.14)—in contrast to the corresponding coefficients at the *δ* potential, given by equation (2.5)—do *not* depend on the propagation constant, *k*.

If we take a negative *δ*′ potential and replace equation (2.6) by
2.15
one now gets, in the anti-symmetric case,
2.16
for , and
2.17a
and
2.17b
for .

Finally, we note that anti-symmetric regularization of the scattering problem for the linear combination
2.18
does not change the scattering coefficients, equations (2.12) and (2.14) for the upper sign in equation (2.18), and equations (2.16) and (2.17) for the lower sign. This is owing to the fact that the singularity of the regularization of the *δ*′ function, given by equations (2.8) and (2.9), will dominate over the regularization of the *δ* function, to be given by equation (4.1).

## 3. Relativistic *δ* scattering problems

The Dirac equation for a particle of mass *m* in one spatial dimension is given by Coutinho *et al.* [9],
3.1
Here, *H* denotes the Hamilton operator, *p* is the momentum operator , *m* is the particle mass and *ψ* is a two-component wave function
3.2
For the Dirac matrices, *α* and *β*, any two of the Pauli matrices
3.3
may be used. The potential, *U*, is composed as
3.4
where *I* is the identity matrix
3.5
As short-range potential components, we shall use
3.6
where *v*,*u*_{x},*u*_{y} and *u*_{z} denote the amplitudes of the *δ* functions. In the non-relativistic limit where the light velocity , the energy, *E*, becomes
3.7

We have selected the six choices indicated in table 1 of the Dirac matrices, *α* and *β*, and the single non-zero components of the potential *U*, which—by differentiation with respect to *x*—yields two uncoupled Schrödinger equations for the components, *ψ*_{a} and *ψ*_{b}, of the wave function to be given approximately in equations (5.1)–(5.5).

From equation (3.1) with (3.2)–(3.7), we thus obtain the following six sets of coupled first-order equations:
3.8a
3.8b
3.9a
3.9b
3.10a
3.10b
3.11a
3.11b
3.12a
3.12b
3.13a
and
3.13b
where we have introduced the propagation constants
3.14
with the corresponding Compton wavelength being *λ*_{C}=*h*/*mc*, and
3.15
for convenience.

We see that equations (3.10) and (3.13) are of one type (denoted type I in §4), while equations (3.8) and (3.9) and equations (3.11) and (3.12) are of another type (denoted type II in §4). In the following section, we shall solve one example of each of the two types. Solutions for the other equations may be obtained by appropriate substitutions of symbols.

## 4. Solution of the Dirac equations by regularization

As the Dirac equations (3.8)–(3.13) cannot be solved by integration, the *δ* functions must be regularized, e.g. by using
4.1
We note that the sequence of functions in **Δ**(*x*) may be arbitrarily chosen as long as the area under the function is unity during the limiting process. However, if different regularizations of the delta functions in the upper and lower equations in the coupled sets of equations (3.8)–(3.13) are chosen, the solution will depend on the regularization. This result, which corresponds to the dependence on the regularization of the solution for the *δ*′ potential, demonstrated in §2 for asymmetric regularization, has *not* been emphasized in earlier work [9,11].

### (a) Type I

As an example of type I, we solve equation (3.10), regularizing *δ*(*x*) by **Δ**(*x*) (4.1). For |*x*|>*ϵ*, the incident, reflected and transmitted fields may be written as
4.2a
and
4.2b
with
4.3
where *k*_{0} and *k*_{2} are given by equation (3.15).

Following O. Penrose (2005, private communication), we introduce the stretched variable, *ξ*=*x*/*ϵ*, into equation (3.8), with *δ*(*x*) regularized by equation (4.1), and obtain
4.4
for |*ξ*|<1 in the limit . The solutions to equation (4.4) can be written
4.5
where *C*_{a} and *C*_{b} are integration constants.

Requiring continuity of *ψ*_{a} and *ψ*_{b} at *x*=−*ϵ* and *ϵ*, i.e. *ξ*=−1 and 1, yields four conditions for four unknown coefficients *R*,*T*,*C*_{a} and *C*_{b}. As a result, the reflection and transmission coefficients become
4.6a
and
4.6b

If we want to write *ψ*_{b} in the form of equation (2.4), we must normalize equation (4.2b) by dividing through by i/*κ*, giving
4.7
with
4.8a
and
4.8b
instead of equation (4.2b) with *R* and *T* given by equation (4.6).

### (b) Type II

As an example of type II, we solve equation (3.8), again regularizing *δ*(*x*) by **Δ**(*x*) (4.1). For |*x*|>*ϵ*, the incident, reflected and transmitted fields may now be written as
4.9a
and
4.9b
with
4.10
where *k*_{1} and *k*_{C} are given by equation (3.14).

Note the relation 4.11 following from equations (3.14) and (3.15).

Proceeding in the same manner as in §4*a*, we get in the limit , for |*ξ*|<1, the equations
4.12
and their corresponding solutions
4.13
Determining the integration constants, *D*_{a} and *D*_{b}, as well as *R* and *T*, from the same continuity requirements as in §4a, we find
4.14a
and
4.14b
where
4.15
has been introduced for convenience.

Writing *ψ*_{b} in the form of equation (2.4), we now normalize equation (4.9b), dividing by (i*K*−*k*_{C})/i*k*_{1}, and obtain
4.16
with
4.17a
and
4.17b
where *R* and *T* are given by equation (4.14).

In the limiting case , to be considered in §5, it follows from equations (4.15), (4.10) and (3.14) that . In this case, equation (4.17) yields 4.18a and 4.18b

In the formal limit , equations (4.15), (4.10) and (3.14) yield . In this case, equation (4.17) gives 4.19a and 4.19b with 4.20a and 4.20b obtained from equation (4.14).

Note that the coefficients in equations (4.19) and (4.20) get the same structure as in equation (4.8). As we shall see in §5, this agrees with the fact that the corresponding Schrödinger equations of types II and I get similar form for .

## 5. On the non-relativistic limit

Differentiating the Dirac equations (3.8)–(3.13) with respect to *x*, we obtain uncoupled Schrödinger equations for the field components, *ψ*_{a} and *ψ*_{b}. Terms of order (1/*c*)^{2} turn out to contain products of *δ* functions. Since the *δ* function is a generalized function, such terms are not meaningful. As a consequence, our Schrödinger equations must be truncated at order 1/*c* in the non-relativistic limit . In this limit, terms beyond order 1/*c* are thus beyond the accuracy of the approximation. Using a perturbative approach based on decoupling techniques (e.g. [14]), this limitation could be overcome since the resulting hierarchy of equations does not include products of potentials.

The approximative equations become 5.1 5.2 and 5.3 corresponding to equations (3.8)–(3.10), respectively, 5.4 corresponding to equations (3.11) and (3.12), and 5.5 corresponding to equation (3.13).

In the following sections, we shall consider equations (5.3) and (5.1) as illustrative examples corresponding to types I and II, respectively, chosen in §4.

### (a) Type I

It is seen that equation (5.3) for *ψ*_{a}(*x*) and *ψ*_{b}(*x*) is of the same form as equation (2.1) in §2 with *U*(*x*) given by the *δ*′ potential equations (2.15) and (2.6), respectively, and . From this comparison,
5.6
with
5.7a
and
5.7b
using equation (2.17), and
5.8
with
5.9a
and
5.9b
using equation (2.14).

The use of equations (2.17) and (2.14) is justified because the condition is fulfilled to order 1/*c* for . Using equations (4.3) and (3.15), we see that *K*=*k*(1+*O*(1/*c*^{2})) in this limit. Furthermore, noting that and to order 1/*c*, we conclude that our solutions to the Dirac equations (3.10), *ψ*_{a} and *ψ*_{b,norm} given by equation (4.2a) with (4.6), and by equation (4.7) with (4.8), respectively, agree with the solution to the approximative Schrödinger equation (5.3), given by equations (5.6), (5.7) and equations (5.8) and (5.9), respectively, within the accuracy of the approximation.

### (b) Type II

Omitting the terms in the limit in equation (5.1), we get an equation of the same form as equation (2.1) in §2, with *U*(*x*) given by the *δ* potential (2.2), with . From this comparison, for both field components, *ψ*_{a}(*x*) and *ψ*_{b}(*x*),
5.10
with
5.11a
and
5.11b
using equation (2.5) with .

For , equations (4.14) and (4.18) give and . Thus, agreement between the solutions to the Dirac equation (3.8), *ψ*_{a} and *ψ*_{b,norm} given by equation (4.9a) with (4.14) and (4.15), and by equation (4.16) with (4.18), respectively, and the solution to the approximative Schrödinger equation (5.1), given by equations (5.10) and (5.11), has been demonstrated in this limit.

Keeping the terms in equation (5.1), we get approximative equations of the same form as equation (2.1), with *U*(*x*) given by equation (2.18), with and .

As noted in §2, the solution to the regularized problem is unaffected by the presence of the *δ* term in the coefficient and thus becomes, with similar accuracy as in §5*a*,
5.12
with
5.13a
and
5.13b
using equation (2.14), and
5.14
with
5.15a
and
5.15b
using equation (2.17).

Neglecting the term *δ* in equation (2.18), i.e. letting , we may compare *R*_{a} and *T*_{a}, given by equation (5.13), to *R* and *T* in equation (4.9a), with *R*=*R*_{π/2} and *T*=*T*_{π/2}, given by equation (4.20), and *R*_{b} and *T*_{b}, given by equation (5.15), to *R*_{norm} and *T*_{norm} in equation (4.16), with *R*_{norm}=−*R*_{π/2} and *T*_{norm}=*T*_{π/2}. As in §5*a*, there is agreement to order 1/*c*, and we may conclude that the solutions to the Dirac equation (3.8) agree with the solutions to the approximative Schrödinger equation (5.1) within the accuracy of the approximation.

## 6. Conclusions

By proper choice of Dirac matrices and single potential components, we have selected the six Dirac-equation scattering problems that lead to uncoupled Schrödinger equations for the two field components in the one-dimensional case.

In the case of short-range potentials, represented by the Dirac *δ* function, we observe, in the non-relativistic limit , that the Schrödinger equations must be truncated at order 1/*c* to avoid terms with products of generalized functions in the equations. Therefore, these Schrödinger equations can only be given in approximate form.

We find two types of Dirac equations (and their corresponding approximative Schrödinger equations) and investigate one example of each type in detail.

The relativistic scattering problem for the *δ* potential cannot be integrated, and must, therefore, be solved by regularization. For this purpose, we use the efficient perturbation techniques based on a stretched coordinate, suggested to us by O. Penrose (2005, private communication).

We point out that arbitrary regularizing functions may be used as long as the area condition is met and the same regularization is used for both delta functions occurring in the two first-order equations. Asymmetric regularization leads to dependence on the regularizing functions.

The non-uniqueness is demonstrated for asymmetric regularization of the corresponding *δ*′ problem for the Schrödinger equation. Here, we use the suggested perturbation techniques and rederive the essential feature that this short-range potential is partially penetrable for a discrete set of values of the potential amplitude.

Table 2 contains a list of the problems treated in the paper.

Differences and ratios of the field components before and after the short-range potential (i.e. at *x*=0− and 0+) are included in table 2. For the Dirac equations, the ratios become simple exponential functions of the amplitudes of the scattering potential.

A comparison between the solutions of the Dirac equation and the corresponding approximative Schrödinger equation is carried out in detail for two types of solutions, denoted types I and II in the paper. Agreement to the level of accuracy of the approximation is found.

In general, the regularization procedure for the scattering problems considered here involves two limiting processes: the regularization parameter and the light velocity .

In the present paper, we first let , leading to generalized functions, *δ* and *δ*′. Then, we consider some consequences of the second limiting process, .

Different strategies—such as to let and at the same time with the product *ϵc* remaining constant during the process, or to take other limiting processes into account—are left open for possible future research.

## Acknowledgements

O. Penrose (2005, private communication) is acknowledged for his seminal suggestion to obtain the regularized solution to the *δ*′ scattering problem by introducing a stretched variable. One of the authors (Y.B.G.) thanks the Department of Informatics and Department of Physics, Technical University of Denmark for hospitality and acknowledges financial support from the Special Programme of the Department of Physics and Astronomy of the National Academy of Sciences of Ukraine and from Civilingeniør Frederik Leth Christiansens Almennyttige Fond.

## Footnotes

This paper is dedicated to the memory of Robin Bullough. He made many original contributions to soliton theory and promoted its applications in optics, statistical mechanics and other fields.

One contribution of 13 to a Theme Issue ‘Nonlinear phenomena, optical and quantum solitons’.

- This journal is © 2011 The Royal Society