The J-Matrix Method Episode 4 docx

Abdelmonem Abstract We describe an efficient and accurate scheme to compute the S-matrix elements for a given multi-channel analytic and non-analytic potentials in complscaled orthonorma

50 40

M = 2

30 20

6.0 4.0

5.0 4.0

3.0 2.0

Fig 2 Same as Fig 1, except that the potential is now two-term separable with V00= 5, V11 = 3,

and V01= V10 = −1

50 40

30 20

10 0

6.0 4.0

2.0 0

3.0 2.0

1.0 0

M = 4

50 40

30 20

10 0

6.0 4.0

2.0 0

0.0

5.0 4.0

3.0 2.0

1.0 0

Fig 4 Same as Fig 1, except that the potential is now four-term separable with the potential

parameters V nm given by (28) and the phase shift is obtained numerically using the relativistic

J -matrix method [2–4, 6]

Acknowledgments I am grateful to Prof H A Yamani for fruitful discussions and suggestions

for improvements on the original manuscript.

References

1 A D Alhaidari, J Phys A: Math Gen 34, 11273 (2001); 37, 8911 (2004)

2 A D Alhaidari, J Math Phys 43, 1129 (2002)

3 P Horodecki, Phys Rev A 62, 052716 (2000)

4 A D Alhaidari, H A Yamani, and M S Abdelmonem, Phys Rev A 63, 062708 (2001)

5 W Magnus, F Oberhettinger, and R P Soni, Formulas and Theorems for the Special tions of Mathematical Physics, 3rd edition (Springer-Verlag, New York, 1966) pp 239–249

Func-6 A D Alhaidari, H Bahlouli, A Al-Hasan, and M S Abdelmonem, Phys Rev A 75, 062711

(2007)

7 H A Yamani and L Fishman, J Math Phys 16, 410 (1975)

for Multi-Channel Analytic and Non-Analytic

H.A Yamani and M.S Abdelmonem

Abstract We describe an efficient and accurate scheme to compute the S-matrix

elements for a given multi-channel analytic and non-analytic potentials in

complscaled orthonormal Laguerre or oscillator bases using the J -matrix method As

ex-amples of the utilization of the scheme, we evaluate the cross section of two-channelsquare wells in an oscillator basis and find the resonance position for the same po-tential using the Laguerre basis We also find resonance positions of a two-channel

Laguerre basis

The work reported here is a continuation of the authors’ effort [1–7] to utilize the

advantages of the J -matrix method of scattering In particular, we have provided a

solution for the Lippman–Schwinger equation in a closed form for a model

finite-rank potential This yields a T -matrix and S-matrix in terms of certain coefficients

evaluated in the basis set chosen In this chapter, we answer the question of how

to accurately and easily evaluate the S-matrix elements for such a potential from

analytic and non-analytic general potentials

This chapter is organized as follows In Sect 2, we review the defining

state-ments of the various functions that come into play in the calculation of the S-matrix

elements In Sect 3, we explicitly outline the various methods available for thecalculation of these functions It is important to be certain of the accuracy of the

desired features It is easy to implement, makes full use of the nature of the basis,and works for general potential Furthermore, its accuracy is assured, even for high

n and m indices, through the proper choice of a parameter K , which specifies the

order of the quadrature scheme utilized We plan to achieve the goal of accurately

calculating the matrix elements V nm in stages In Sect 3.1 we propose an imate scheme to determine the matrix elements of a one-channel central analytic

this scheme does not work for non-analytic potentials, in Sect 3.2 we develop anapproximate scheme that may be applied for both kinds of potentials In Sect 3.3and 3.4 we generalize the scheme in the preceding two sub-sections to find the

and have the form

λ α = |λα| e −iθ α , and λ β=λβe −iθ β

The results of Sect 3 that are associated with the use of the oscillator basis aresummarized in Sect 3.5

Finally, in Sect 4 we discuss the accuracy of the proposed scheme and illustrateits application by considering two examples In the first we evaluate the cross section

of two-channel square wells in the oscillator basis, and find the complex resonanceenergy of the same potential using the Laguerre basis In the second example wefind the complex resonance energy for a specific two-channel analytical potential

consider the same problem with the Coulomb term included in the calculation usingthe Laguerre basis

1 Review of the J -Matrix Method

Here we consider the scattering of a structureless, spinless particle by a target with

M internal states labeled by the threshold energies E1, E2, , E M We assume that

H0= −12

or the oscillator basis

n(ζ ) is the generalized Laguerre

momentum quantum number

The J -matrix method of scattering finds the exact S-matrix to the model

po-tential, V , by restricting its infinite matrix representation in a complete L2 basis to

a finite representation That is, the model potential ˜V is defined as



=φ(α)

m  ¯φ(α) n

Y α,β = −g(α,β)

N α −1,N β−1 J N(β) β −1,N β R+N

and in (8) we show the first row and column, while the remainder is represented by

⌬( +)(E) of equation (7) Additionally, R±

N(α) and T N−1(α) are the J-matrix ratios



is the set of coefficients of the expansion of the

Hamiltonian matrix The matrix element g(N α,β) α −1,N β−1 takes the following explicitforms:

(i) In the Laguerre basis

α=1

the scattering Hamiltonian

Thus, the calculation of the exact S(E) for the model potential requires:

(2) The matrix element g(N α,β) α −1,N β−1(E).

Both Bases

We first find expressions for the matrix elements of the J matrix, which are

sym-metric and tridiagonal in both bases

(1) In the Laguerre basis:

La-guerre or oscillator bases In order to do that, we first define the following functions

calcu-late R±1 (E) and T0(E), where

R±1 = (c1± is1)

(c0± is0) and T0= (c0− is0)

presented as continued fraction expressions elsewhere [9]

We may then determine the matrix elements of the reference Hamiltonian in

Re(λ) > 0 (Laguerre) and Re(λ2) > 0 (oscillator) If we write λ = |λ| e −iθ, the

φ θ

nH0φ θ m

are calculatedonce, and used with every model potential that is considered The matrix elements

(i) For the Laguerre basis:

(smaller) of the two indices n and m, respectively.

(ii) For the oscillator basis (z= 0 only):

3.1 The Case of a Single-Channel Central Analytic Potential

reli-n areli-nd m If the scale parameter of the basis is real, the above matrix elemereli-nt V nm

may be cast in the form

It is known [10, 11] from the theory of orthogonal polynomials that associated with

single element of unity in the nth row Thus, the potential matrix integral (31) can

Computationally, this is an extremely simple formula Performing the

k=0 The summand in equation (38) is the

value of the potential at the scaled positions



ζ (K )

of the nth and mth coefficient of the eigenvectorv ( K )

k

&

Since the argument of thepotential in the sum (38) is real, this approximation works just as well for bothanalytic and non-analytic central potentials

The corresponding result of equation (38) for a complex scale parameter having

definition (26) for the matrix elements yields an integral similar to equation (27),

theo-rem shows that the integral along the stated ray is equivalent to the integral (27)

The evaluation of the matrix elements of one-channel potentials in the oscillator

result corresponding to equation (38) may be shown by the equation

the form

3.2 The Case of a Single-Channel Central Non-Analytic Potential

We generalize the results in the previous subsection to the case of non-analytic tential We start with the Laguerre basis in which the scale parameter is complex

the integral (31) can be cast in the form

k=0, we are able to obtain

an approximation to the above integral, as

The following points may be noted concerning the above expression First, it makesuse of the eigenvalues and eigenvectors of the finite tridiagonal matrix of equa-

tion (34) Second, the ratio p n

Finally, the potential is sampled at the real points

<

ζ (K )

k / (|λ| cos θ)=K−1

the result (46) applicable to both analytic and non-analytic potentials

The evaluation of the matrix elements of one-channel potentials in the oscillator

in a fashion similar to that of the above Laguerre case In fact, the equation for the

3.3 The Case of a Multi-Channel Central Analytic Potential

We generalize the results obtained in the last section to the case of a multi-channel

all channel bases to be of the Laguerre form (3) and we allow the channel scaleparameterλ αto differ fromλ β The matrix element V nm αβ is now given by

A consideration ofλ αβ = λ α + λβ/2 with λ α = |λα| e −iθ α andλ β = λβe −iθ β

results in the restriction 0≤ θα , θ β < π/2 We note that the integral in equation (54)

replaced by the product F nm αβ(ζ ) V αβ

ζ/λ αβ Therefore, the technique we used toapproximate the integral of equation (33) may be used to approximate the integral

of equation (54) Thus, the corresponding multi-channel result in the Laguerre basis(3) is given by

reduces to that of the one-channel case (46)

As has been noticed in the one-channel case, the similarities and differences tween the evaluation of the one-channel potential matrix elements in the Laguerreand oscillator bases are preserved in the multi-channel case Thus, the matrix ele-

/2 with λ α = |λ α | e −iθ α, andλ β = λβe −iθ β The

multi-channel results in the oscillator basis (4) are given by

3.4 The Case of a Multi-Channel Central Non-Analytic Potential

The above results reduce to the one-channel result of the previous section when

the oscillator basis (4), is given by

4 Discussion and Numerical Examples

The scheme employed to find an approximate value of the potential matrix element

where f ( ζ ) = p n(ζ ) p m(ζ ) V (ζ/λ) /⌫ (ν + 1) and R N is the remainder The

we consider a general potential, it is not possible to obtain a tighter bound on theremainder than the bound already available from the classical treatment of the Gaussquadrature scheme associated with the theory of orthogonal polynomials [12] In-stead, we prefer to illustrate the scheme by evaluating specific examples

Example 1 Scattering and Resonance Information

for a Two-Channel Square Well Potential

The method of complex scaling has proven to be very powerful in locating nances [1] in the complex energy plane as well as in carrying out calculations ofmulti-channel scattering [2] It involves the evaluation of a given system Hamilto-

reso-nian, H , in a finite rotated basis, φ θ

n

N

n=0, or alternatively [3, 4] the evaluation

We propose to test the method developed in the previous section by applying it

in the calculation of s-wave scattering for the two-channel square well problem aspreviously considered by Rescigno and Reinhardt [2]

V αβ (r ) = V0αβ r ≤ 1

= 0 r > 1.

potential strengths are taken to be

cho-sen The matrix elements of the reference Hamiltonian are given by equation (25)

We also chose the order of approximation K to equal 60 Figure 1 plots the results

for the un-normalized elastic cross sectionsπ |1 − S11|2 andπ |1 − S22|2, and the

un-normalized exact cross sections that can be obtained analytically for this problem[13] Furthermore, we find the position of the resonance of the same potential using

approximation K to equal 60 We then evaluate the matrix elements of the potential,

Fig 1 Un-normalized elastic and inelastic cross sections for s-wave scattering from two coupled

square wells Potential parameters are those of equation (77) and the potential matrix elements are calculated using equation (72) Exact results for the un-normalized elastic cross section for chan- nels 1 and 2 are indicated as solid and dashed lines respectively; for the un-normalized inelastic

cross section are shown as dashed-dotted lines The order of approximation K is taken as 60 The

basis set consisted of 30 oscillator functions for each channel The free parameterλ1= λ2= 1.0

was used for both channels, as was a rotation angle ofθ1 = θ2 = 0.0 radians The calculated

un-normalized elastic cross sections for channels 1 and 2 are indicated by open circles and squares respectively, while the un-normalized inelastic cross sections are indicated by solid circles

increasing N , as shown in Table 1.

Table 1 The position of resonance for s-wave scattering from two coupled square wells of

equa-tion (77) as a funcequa-tion of basis-set size N in comparison with the exact result The order of imation K is taken as 60 We use the Laguerre basis with a complex scale parameter λ = |λ| e −iθ.

approx-Here,|λ1| = |λ2| = 6.0, rotational angles θ1= θ2= 0.1rad

Example 2 Resonance Information for a Two-Channel

Analytic Potential

In this example, we find the complex resonance energy for a specific two-channel

both bases We then consider the same potential when the Coulomb term is includedusing the Laguerre basis The matrix elements of the reference Hamiltonian aregiven by equation (24) We apply the proposed method to the characterization of

the s-wave narrow resonance for a two-channel problem for a case in which z = 0

while using both the Laguerre and oscillator bases, a model two-channel problempreviously considered by Noro and Taylor [14] and Mandelshtam et al [15] The

problem consists of the scattering of a structureless particle with a charge of z = 0

by a target that has only two internal states with threshold energies 0.0 and 0.1 a.u.The matrix elements of the interaction potential are taken to be

considered the same and equal 5.0 for the Laguerre basis and 1.5 for the oscillator

order of approximation K equal 60 We then evaluate the matrix elements of the

different values of N , up to N = 50 for each channel A stable resonance state for

 = 0 is found, whose energy converges to the value ε = 4.7682 − i 0.000710 with increasing N , as shown in Table 2 This result compares well with that of Mandelsh-

tam et al [15] We notice that the accuracy of the results using the oscillator basis

is a little better than that observed when using the Laguerre basis The calculations

that the calculations converge quickly with increasing N for both the oscillator and

Laguerre bases Furthermore, the accuracy of calculations using the oscillator basis

is similar to the accuracy of calculations using the Laguerre basis

which z = +1.0 and –1.0 within a Laguerre basis, which has the advantage of being

Table 3

The variety of examples presented in this chapter show that the Gauss quadraturescheme is a natural and accurate way of evaluating potential matrix elements incomplex scaled Laguerre and oscillator bases

Table 2 Results for the two-channel potential of equation (78) as a function of basis-set size N for

different angular momenta and z = 0 The order of approximation K is taken as 60 We use bases

with a complex scale parameterλ = |λ| e i θ Here,θ1 = θ2 = 0.1rad., |λ1| = |λ2| = 1.5 for the

oscillator basis, and|λ1| = |λ2| = 5.0 for the Laguerre basis The potential matrix elements are

calculated using equation (56) for the Laguerre case and equation (59) for the oscillator case The result of the calculation using = 0 is compared with that of Mandelshtam et al [15]

Table 3 Results for s-wave scattering of the two-channel potential of equation (78) as a function of

basis-set size N with z = ±1 The order of approximation K is taken as 60 We use the Laguerre

basis with a complex scale parameterλ = |λ| e −iθ Here,|λ1| = |λ2| = 5.0, rotational angles

θ1= θ2= 0.5rad The potential matrix elements are calculated using equation (56)

1 For a review of this method see, Reinhardt W P 1982 Ann Rev Phys Chem 33, 223

2 Rescigno T N and Reinhardt W P 1973 Phys Rev A8, 2828

3 Rescigno T N and McCurdy C W 1986 Phys Rev A34 1882

4 Yamani H A and Abdelmonem M S 1996 J Phys A: Math Gen 29, 6991

5 Arickx F, Broeckhove J, Van Leuven P, Vasilevsky V, and Filippov 1994 Am J Phys 62, 362

6 Alhaidari A D, Bahlouli H, Abdelmonem M S, Al-Ameen F, and Al-Abdulaal T (2007) Phys.

Lett A 364, 372