Relativistic J-matrix method of scattering in 1 + 1 space-time I. Theory

We make a relativistic extension of the one-dimensional J-matrix method of scattering. The relativistic potential matrix is a combination of vector, scalar, and pseudo-scalar components. These are non-singular short-range potential functions (not necessarily analytic like a square well) such that they are well represented by their matrix elements in a finite subset of a square integrable basis set. This set is chosen to support a tridiagonal symmetric matrix representation for the free Dirac operator. We derive the transmission and reflection coefficients. This work will be followed by another where we apply the theory to obtain scattering information for different potential coupling modes including those with evidence of Klein tunneling and of supercritical resonances.


Introduction
Not long ago, we presented an algebraic formulation of the nonrelativistic scattering problem in one dimension based on the J-matrix method [1]. We found that the usual thought of simplicity of the 1D scattering problem, when compared to 3D, is not manifest as indicated by the nontrivial and highly rich structure that emerged. For example, there will always be two scattering channels that are non-trivially coupled. The two channels decouple only if the interaction potential has a definite parity (i.e., being either odd or even function in the configuration space coordinate x). In that formulation, we deployed the analytic power built in the J-matrix method where the theory of orthogonal polynomials plays an important role [2][3][4][5][6][7][8]. Additionally, we exploited the efficiency of numerical schemes associated with tridiagonal matrices that resulted in a convergent and accurate calculation of the reflection and transmission amplitudes. These schemes include Gauss quadrature and continued fraction techniques [9][10][11][12][13]. We have also demonstrated that the formulation could be used as an alternative means for resolving issues related to bound states, resonances and scattering phenomena in one-dimensional systems. Moreover, we have shown that it constitutes a viable alternative to the classical treatment of the 1D scattering problems and that it could also help unveil new and interesting applications. We believe that the relativistic extension of that theory will have similar qualities. Therefore, the objective here is to present a relativistic extension of the work in [1]. That is, we would like to formulate the relativistic J-matrix method of scattering in 1+1 Minkowski space-time. This theory becomes relevant when studying one-dimensional models at higher energies compared to the rest mass energy or at strong coupling. Moreover, it offers various choices of coupling modes for the same potential configuration. These modes include vector, scalar, pseudo-scalar, and a mix of these. The reader is strongly advised to consult [1] for background, introduction and notation.
Consequently, we are interested here in the scattering solution of the following time-independent Dirac equation in 1+1 Minkowski space-time y e y + ñ= ñ H , 1 where ψ is a two-component wavefunction and ε is the relativistic energy. H 0 is the free Dirac operator and ν is a 2×2 potential matrix. In the relativistic units = =  c 1and in a typical representation of the Dirac gamma matrices in 1+1 space-time, where we can take g s = = -1 0 0 1 0 3 ( ) and g s = = i i 0 1 1 0 , 1 1 ( ) we can write 1 = - where M is the rest mass of the Dirac particle. The two-vector potential is Since e = +M (e = -M ) belongs to the positive (negative) energy spectrum, then the positive energy subspace of interest to this work corresponds to the top signs and left vertical arrows. Consequently, we obtain the following two independent positive energy solutions for the reference Dirac equation , M M A and  are arbitrary real constants. In the terminology of the J-matrix method, we refer to these as the sine-like solutions. However, as in the non-relativistic 1D J-matrix [1], we end up with two scattering channels: an even channel and an odd channel to which + S x ( ) and -S x ( ) belong, respectively. Due to the finite range of the potential, the boundary conditions for the scattering problem, which is defined by the positive energy solution of equation (1) These represent a normalized flux of relativistic particles with energy ε incident on the potential region from left that gets partially reflected with an amplitude e R ( ) and partially transmitted with an amplitude e T .
( ) These amplitudes depend not only on the energy but also on the potential parameters. Unitarity of the problem results in the current (particle flux) conservation equation, + are required to carry a tridiagonal matrix representation for the reference wave operator, x, λ is a real positive scale parameter of inverse length dimension, H y n ( ) is the Hermite polynomial of order n, and the normalization constants is = A n p n 2 2 .
On the other hand, the spinor basis for the odd channel is obtained from that of the even channel (9) as follows for all integers n and m. Moreover, using the orthogonality relation of the Hermite polynomials [14,15] and the integral formula in [16], we obtain the following energy-dependent expansion coefficients for the even and odd channels On the other hand, the matrix elements in the odd subspace, f f º á ñ | | is obtained from those in (14a) by the replacement  + n n Since the reference wave operator is an even matrix function 2 , ( ) 2 An even/odd 2×2 matrix function Q(x) must have its two diagonal elements Q 11 (x) and Q 22 (x) as even/odd functions of x and its two offdiagonal elements Q 12 (x) and Q 21 (x) as odd/even functions of x.
The recurrence relation for e s n ( ) is obtained from (15) by the replacement  + n n .
Now, the scattering information for the even and odd channels is contained in the difference at infinity between the phases of  S x ( ) and those of the complementary solutions  C x , ( ) which will be obtained next. This phase difference depends on the energy and potential parameters. In the absence of interaction ( = 0 V ) it is p 2 for all energies. The complementary solutions (known in the J-matrix language as the cosine-like solutions) are written as (1) Be a second independent solution of the energy differential equation (13).
(2) Be a second independent solution of the three-term recursion relation (15a) and (16a) with initial relations that differ from (15b) and (16b).
(3) The initial relation for the recursion of e  c n ( ) is chosen such that it makes the complementary reference wavefunctions  C x ( ) asymptotically sinusoidal and identical to  S x ( ) but with a phase difference of p . 2 These conditions result in the following expansion coefficients for where F a c z ; ; 1 1 ( ) is the confluent hypergeometric function. For the details of these calculations, one may consult [1]. The contiguous relation of F a c z ; ; could be used to verify that (17a) and (17b) satisfy the three-term recursion relation (15a) and (16a) for = n 1, 2, 3,..., respectively. On the other hand, the initial relations that replace (15b) and (16b) are, respectively, as follows A simple argument in support of these findings is the observation that, in one of its representations in terms of the confluent hypergeometric function, the Hermite polynomial could be written as follows . 21 n n n n n n 1 2

Therefore, for even degrees this relation gives
in which case only the e + s n ( ) term survives since G -n ( )blows up. Similarly, for odd degrees the same relation gives m e -µ - in which case only the e s n ( ) survives. Now, the cosine-like solutions have the following useful asymptotic properties We have shown elsewhere [18] that if we eliminate the lowest terms in the series and construct the truncated series ) for some large enough integer N, then we obtain vanishingly small values for these functions within a finite region symmetric about the origin. The size of this region increases with the number of eliminated terms, N [18]. In fact, there is a perfect parallel between the limit as  ¥ N of these functions and the limit as As we shall see below, this property will proof very valuable in the construction of the complete solution of the problem such that it satisfies the asymptotic boundary conditions (7a) and (7b). Next, we will augment the kinematics obtained above by the dynamics coming from the interaction of the Dirac particle with the short range potential matrix x V( ) and identify the scattering amplitudes R(ε) and T(ε).

Interaction and the scattering amplitudes
The potential matrix x V( ) is short-range such that it is well represented by its matrix elements in a large enough subset of a square integrable spinor basis set, x x .
The size of the middle subspace, whose dimension is 2N, should be large enough to give an accurate matrix representation of the short-range scattering potential. Moreover, it should be large enough to give a faithful representation for the asymptotic sinusoidal limit of the total wavefunction as  ¥ x .

| |
Using the asymptotic properties of the sine-like and cosine-like solutions given by equation (23), we can write the boundary conditions for large enough N. Comparing these asymptotic solutions represented as sums with the corresponding sum in equation (25) we deduce the following We also require that the subset of the basis in the middle subspace be energy independent so that numerical computations (e.g., of the potential matrix elements, Harris eigenvalues, matrix diagonalization, etc) will be done once for all energies. Moreover, to make the numerical scheme very effective and simple, we maintain the tridiagonal structure of the matrix representation of the reference wave operator in the middle subspace. These requirements allow us to choose the lower components of x where τ is a new dimensionless basis parameter (in addition to λ). Now, to realize the solution of the problem (i.e., give the complete specification of the total wavefunction ψ) we only need to determine the Now, we define the finite 2N×2N matrix Green function as given by equations (28a)-(28d) is orthogonal, then the components of this matrix Green function is calculated as shown in appendix A by using either formula (A12) or (A17). Inverting the matrix equation (30) results in the following two equations and the following quantities

{ }
We also show how to obtain an accurate evaluation of the matrix elements of the short-range scattering potential n m , V by using Gauss quadrature integral approximation.

Conclusion
In this paper, we presented a theory of the relativistic J-matrix method of scattering in 1+1 flat space-time. It turned out to be a non-trivial extension of the non-relativistic theory, which was formulated by our group in [1]. We will follow this work with another that apply the theory to specific relativistic models in one dimension as demonstration of the utility and accuracy of this formulation. As in the standard J-matrix method, it is important to note that the physical results obtained will be independent of the values of the basis parameters λ and τ as long as these values are within the plateau of stable computations. A calculation strategy to find the plateau of stability for computational parameters is given in [19]. Moreover, the size of the plateau of stability increases with N and, in principle, as  ¥ N the results should be independent of any choice of values for these parameters.
In these future studies, we plan to treat the relativistic scattering problem for various potential configurations and coupling modes (vector, scalar, and/or pseudo-scalar). We will also consider spin symmetric where r =  [20]. However, we have been utilizing it since the early inception of the J-matrix method.
( ) On the other hand, to calculate the matrix elements of the short-range potential, =