The J-Matrix Method Episode 2 docx

16 E.J. Heller, H.A. Yamani
concerning H
0
that can be extracted decreases with the number of basis functions
per channel, and this can cause difficulties when basis size is a restriction. On the
other hand, approximate treatment of H
0
can be advantageous when for example
channel threshold details are of no interest or are unwanted artifacts of particular
models. In the J-matrix method, H
0
is accounted for exactly independent of basis
size. Thus we start with a large part of the problem “diagonalized” and the full
analytic structure of the S matrix is built into the problem, raising the hope that
quite small basis sets will be sufficient for many problems. The analytic nature of
the solutions allows variational corrections to be made and provides a solid footing
for further theoretical work.
We now summarize the steps necessary to perform a calculation with the J -matrix
method. First, the potential V
N
(or
˜
V ) is evaluated in the Laguerre basis set; and is
then added to the N × N tridiagonal representation of H
0
− E. To this inner matrix
we add one extra row and column, for each asymptotic channel, containing matrix
elements of H
0
and the cos(n + 1)θ terms. The right-hand side “driving” terms are
similarly constructed with the sin(n +1)θ terms. The resulting linear equations can

be solved efficiently if a pre-diagonalizing transformation ⌫ is applied to the inner
matrix as in Section 2. If desired, the matrix elements of H
0
+V
N
− E can be eval-
uatedintheSlaterset(λr)
n
e
−λr
/
2
, n = 1, 2, ,N, since these are just transformed
Laguerres. Then a different transformation ⌫

will be necessary to pre-diagonalize
the inner matrix.
In the following chapter we apply the method presented here to s-wave electron-
hydrogen scattering model. The generalization of the method to all partial waves for
both Laguerre and Hermite basis sets has been derived and will be the subject of
a future publication. The case where H
0
, contains the term α
/
r (i.e., the Coulomb
case) is also worked out for Laguerre sets.
Acknowledgments We are grateful for helpful discussions with Professor William P. Reinhardt
and his support of this work. We have also benefited greatly from conversations with L. Fishman,
A. Hazi, T. Murtaugh, and T. Rescigno. This work was supported by a grant from the National
Science Foundation.

References
1. E.J. Heller and H.A. Yamani, following paper, Phys. Rev. A 9, 1209 (1974).
2. T. Kato, Progr. Theor. Phys. 6, 394 (1951).
3. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York,
1965).
4. C. Schwartz, Ann. Phys. (N.Y.) 16, 36 (1961).
5. E.J. Heller, W.P. Reinhardt, and Hashim A. Yamani, J. Comp. Phys. 13, 536 (1973).
6. H. Feshbach, Ann. Phys. (N.Y.) 19, 287 (1962).
7. F.E. Harris, Phys. Rev. Lett. 19, 173 (1967).
8. P.G. Burke, D.F. Gallaher, and S. Geltman, J. Phys. B 2, 1142 (1962).
9. See for example, A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30, 257 (1958); A. M. Lane
and D. Robson, Phys. Rev. 178, 1715 (1969).
New L
2
Approach to Quantum Scattering: Theory 17
10. P.J.A. Buttle, Phys. Rev. 160, 719 (1967); P.G. Burke and W.D. Robb, J. Phys. B 5, 44 (1972);
E.J. Heller, Chem. Phys. Lett. 23, 102 (1973).
11. C. Schwartz, Phys. Rev. 124, 1468 (1961). For a comprehensive review of the algebraic meth-
ods for scattering, including a discussion of pseudo-resonances, see D.G. Truhlar, J. Abdallah,
and R.L. Smith, Advances in Chemical Physics (to be published).
12. R.G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966),
p. 274.
13. W.P. Reinhardt, D.W. Oxtoby, and T.N. Rescigno, Phys. Rev. Lett. 28, 401 (1972);
T.S. Murtaugh and W.P. Reinhardt, J. Chem. Phys. 57, 2129 (1972).
14. E.J. Heller, W.P. Reinhardt, and H.A. Yamani, J. Comp. Phys. 13, 536 (1973); E.J. Heller, T.N.
Rescigno, and W.P. Reinhardt, Phys. Rev. A 8, 2946 (1973); H.A. Yamani and W.P. Reinhardt
Phys. Rev. A11, 1144 (1975).
15. T.N. Rescigno and W.P. Reinhardt, Phys. Rev. A 8, 2828 (1973); J. Nuttall and H. L. Cohen,
Phys. Rev. 188, 1542 (1969).
J-Matrix Method: Extensions to Arbitrary

Angular Momentum and to Coulomb Scattering
Hashim A. Yamani
∗
and Louis Fishman
Abstract The J-matrix method introduced previously for s-wave scattering is
extended to treat the th partial wave kinetic energy and Coulomb Hamiltonians
within the context of square integrable (L
2
), Laguerre (Slater), and oscillator
(Gaussian) basis sets. The determination of the expansion coefficients of the con-
tinuum eigenfunctions in terms of the L
2
basis set is shown to be equivalent to the
solution of a linear second order differential equation with appropriate boundary
conditions, and complete solutions are presented. Physical scattering problems are
approximated by a well-defined model which is then solved exactly. In this manner,
the generalization presented here treats the scattering of particles by neutral and
charged systems. The appropriate formalism for treating many channel problems
where target states of differing angular momentum are coupled is spelled out in
detail. The method involves the evaluation of only L
2
matrix elements and finite
matrix operations, yielding elastic and inelastic scattering information over a con-
tinuous range of energies.
1 Introduction
In two previous publications [1,2] (referred to as I and II) the J-matrix (Jacobi ma-
trix) method was introduced as a new approach for solution of quantum scattering
problems. As discussed in I, the principal characteristics of the method are its use
H.A. Yamani
Ministry of Commerce & Industry, P.O. Box 5729, Riyadh 11127, Saudi Arabia

e-mail:
∗
Supported by a fellowship from the College of Petroleum and Minerals, Dhahran, Saudi Arabia.
Reprinted with permission from: J-matrix Method: Extension to Arbitrary Angular Momentum
and to Coulomb Scattering, H.A. Yamani and L. Fishman, Journal of Mathematical Physics 16,
410–420 (1975). Copyright 1975, American Institute of Physics.
A.D. Alhaidari et al. (eds.), The J-Matrix Method, 19–46. 19
C

Springer Science+Business Media B.V. 2008
20 H.A. Yamani, L. Fishman
of only square integrable (L
2
) basis functions and its ability to yield an exact solu-
tion to a model scattering Hamiltonian, which, in a well-defined and systematically
improvable manner, approximates the actual scattering Hamiltonian. The method is
numerically highly efficient as scattering information is obtained over a continuous
range of energies from a single matrix diagonalization.
The development of the J-matrix method as presented in I is based primarily
upon the observation that the s wave kinetic energy,
H
0
=−
1
2
d
2
dr
2
(1)

can be analytically diagonalized in the Laguerre(Slater) basis:
φ
n
(λr) = (λr)e
−λr
/
2
L
1
n
(λr), n = 0, 1, ,∞, (2)
where λ is a scaling parameter. This follows from the fact that the infinite matrix
representation of

H
0
− k
2

2

in the above basis is tridiagonal (i.e., J or Jacobi
matrix) and that the resulting three-term recursion scheme can be analytically solved
yielding the expansion coefficients of both a “sine-like”
˜
S(r) and a “cosine like”
˜
C(r) function. The J-matrix solutions
˜
S(r)and

˜
C(r) are used to obtain the exact
solution of the model scattering problem defined by approximating the potential V
by its projection V
N
onto the finite subspace spanned by the first N basis functions.
That is, the exact solution ⌿ of the scattering problem,

H
0
+ V
N
− k
2

2

⌿ = 0, (3)
is obtained by determining its expansion coefficients in terms of the basis set
{
φ
n
}
subject to the asymptotic boundary condition,
⌿ →
˜
S(r) + tan δ
˜
C(r), (4)
where δ is the phase shift due to the potential V

N
.
This chapter is intended to generalize the formalism developed in I in three areas.
First, the results of I are extended to all partial waves, in which case the uncoupled
Hamiltonian becomes the th partial wave kinetic energy operator,
H
0
=−
1
2
d
2
dr
2
+
( + 1)
2r
2
, (5)
which has a Jacobi representation in the Laguerre basis,
φ
n
(λr) = (λr)
+1
e
−λr
/
2
L
2+1

n
(λr), n = 0, 1, ,∞, (6)
Secondly, a similar analysis of the Hamiltonian of Eq. (5) is presented within the
context of the oscillator(Gaussian) basis,
J-Matrix Method 21
φ
n
(λr) = (λr)
+1
e
−λ
2
r
2

2
L
+1
/
2
n
(λ
2
r
2
), n = 0, 1, ,∞, (7)
which preserves the Jacobi representation and is also analytically soluble. The third
generalization involves the analysis of the th partial wave Coulomb Hamiltonian,
H
0

=−
1
2
d
2
dr
2
+
( + 1)
2r
2
+
z
r
(8)
in the Laguerre function space of Eq. (6), which again yields a Jacobi form and
is subsequently analytically soluble. It is noted that the analysis of the Coulomb
Hamiltonian in the oscillator set of Eq. (7) does not lead to a Jacobi form.
For the solution of these problems, a general technique is developed which re-
duces the solution of the infinite recurrence problem for the asymptotically “sine-
like” J-matrix eigenfunction to the solution of a linear second order differential
equation with appropriate boundaryconditions. An asymptotically “cosine-like” so-
lution which obeys the same differential equation with different boundary conditions
is then constructed. The fact that both J-matrix solutions obey the same recurrence
scheme is essential to the success of the method as an efficient technique for solving
scattering problems [1].
The program of the chapter is as follows: In Section 2.1, the generalized H
0
problem is considered and a general procedure for obtaining the expansion coeffi-
cients of the sine-like and cosine-like functions in terms of the basis sets is outlined.

In Section 2.2, the general method is illustrated in detail for the case of the radial
kinetic energy in a Laguerre basis. The analogous results for the oscillator basis
and for the Coulomb problem are outlined in Sections 2.3 and 2.4, respectively. The
details of the Coulomb derivation are given in the Appendix. Section 3 contains the
application of the results thus obtained to potential scattering problems. This sec-
tion presents a formula which allows for the computation of phase shifts. Section 4
presents the natural generalization of the J-matrix method to multi-channel scatter-
ing. Finally, Section 5 contains a brief discussion of the over all results and sugges-
tions for applications and areas of further theoretical interest.
2TheH
0
Problem
The problem examined in this section is the “solution” of the equation,

H
0
− k
2

2

⌿ = 0(9)
within the framework of the L
2
function space
{
⌽
n
}
in such a manner as to obtain

both an asymptotically sine-like and asymptotically cosine-like function. The two
J-matrix solutions,
˜
S(r)and
˜
C(r), form the basis for the asymptotic representa-
tion of the scattering wavefunction associated with the full problem. It will also
be required that the expansion coefficients of both
˜
S(r)and
˜
C(r) satisfy the same
three-term recursion scheme,
22 H.A. Yamani, L. Fishman
2.1 Generalized H
0
Problem
The basic differential equation

H
0
− k
2

2

⌿
0
= 0 (10)
possesses both a regular and an irregular solution which behave near the origin as

⌿
0
reg
∼
r→0
r
+1
, (11a)
⌿
0
irreg
∼
r→0
r
−
(11b)
and asymptotically as
⌿
0
reg
∼
r→∞
sin ξ, (12a)
⌿
0
irreg
∼
r→∞
cos ξ, (12b)
where ξ = kr −π


2 in the free particle case and ξ = kr +tln(2kr)−π

2+σ

in
the Coulomb case. In the above, the definitions t =−z

k and σ

= arg⌫(+1−it)
have been used [3].
Since the basis set
{
φ
n
}
is complete for functions regular at the origin, ⌿
0
reg
can
be expanded as
⌿
0
reg
≡
˜
S(r) =
∞


n=0
s
n
φ
n
(λr) (13)
with the expansion coefficients s
n
being formally given by [4]
s
n
=

¯
φ
n
(λr)


˜
S(r)

(14)
with
¯
φ
n
satisfying

¯

φ
n


φ
m

= δ
nm
. A differential equation satisfied by the set of
coefficients
{
s
n
}
can be constructed in the following manner. Since the basis set
{
φ
n
}
tridiagonalizes the operator

H
0
− k
2

2

,the

{
s
n
}
satisfy a three-term recur-
sion relation of the form,
[a
1n
+ a
2n
g(η)] s
n
+ a
3n
s
n−1
+ a
4n
s
n+1
= 0, n > 0, (15a)
[a
10
+ a
20
g(η)] s
0
+ a
40
s

1
= 0, n = 0, (15b)
where η is the energy variable defined by η = k

λ and g(η) is a function dependent
upon the particular choice of
{
φ
n
}
and H
0
. Differentiating Eq. (14) with respect to
x,wherex is a function of the energy variable η appropriate to the particular case,
leads to a differential difference equation of the form
J-Matrix Method 23
η
dx
dη
ds
n
dx
= b
1n
s
n+1
+ b
2n
s
n

+ b
3n
s
n−1
, n > 0 (16a)
η
dx
dη
ds
0
dx
= b
10
s
1
+ b
20
s
0
, n = 0. (16b)
For the case of the Laguerrefunction space, b
2n
= 0, while for the oscillator function
space the general form of Eq. (16) is appropriate. Combining Eqs. (15) and (16)
yields a linear second order differential equation of the form
A(x)
d
2
s
n

dx
2
+ B(x)
ds
n
dx
+ D(x)s
n
= 0 (17)
with two linearly independent solutions χ
1
and χ
2
and a general solution of the form
s
n
= α
1
χ
1
+ α
2
χ
2
. (18)
Equation (15b) determines s
n
to within a normalization constant. The advantage
of the differential equation approach is that a cosine-like solution
˜

C(r), whose ex-
pansion coefficients will also satisfy the differential equation (17), can be readily
constructed.
The cosine-like J-matrix solution,
˜
C(r) =
∞

n=0
c
n
φ
n
(λr), (19)
is constructed to be (1) regular at the origin like ⌿
0
reg
so as to be expandable in
the basis set
{
φ
n
}
, (2) behave asymptotically as ⌿
0
irreg
, and (3) to have its expan-
sion coefficients
{
c

n
}
satisfy Eq. (15b). This immediately means that
˜
C(r) cannot
satisfy the homogeneous differential equation (10). By choosing
˜
C(r) to satisfy the
inhomogeneous differential equation

H
0
− k
2

2

˜
C(r) = β
¯
φ
0
(λr), (20)
the Green’s function [5]
G(r, r

) = 2⌿
0
reg
(r

<
)⌿
0
irreg
(r
>
)

W

⌿
0
reg
, ⌿
0
irreg

(21)
maybeusedtoobtainthesolution[5]
˜
C(r) =−
2β
W

⌿
0
irreg
(r)

r

0
dr

⌿
0
reg
(r

)
¯
φ
0
(λr

) + ⌿
0
reg
(r)

∞
r
dr

⌿
0
irreg
(r

)
¯

φ
0
(λr

)

,
(22)
24 H.A. Yamani, L. Fishman
where W

⌿
0
reg
, ⌿
0
irreg

is the Wronskian of the two independent solutions ⌿
0
reg
and
⌿
0
irreg
and is independentof r and β is a free parameter [6].
˜
C(r) as given by Eq. (22)
is regular at the origin and with the choice
β =−W


2s
0
(23)
goes asymptotically as ⌿
0
irreg
. The fact that the inhomogeneity of Eq. (20) is orthog-
onal to the set
{
φ
n
}
for n = 1, 2, ,∞, implies that, for n > 0, the
{
c
n
}
satisfy
the same three-term recursion relation as the
{
s
n
}
, Eq. (15a). The n = 0 case has the
form
[a
10
+ a
20

g(η)] c
0
+ a
40
c
1
= l(η) = 0 (24)
where l(η) is a function which depends upon the form of β and upon any terms that
were divided out in the derivation of the homogeneous recursion relation, Eq. (15).
Equation (24) is to be contrasted with the homogeneous initial condition
[a
10
+ a
20
g(η)] s
0
+ a
40
s
1
= 0, (15b)
which occurs in the sine-like J-matrix solution. It may be shown from Eqs. (22)
and (23) that the set
{
c
n
}
satisfies a differential difference equation analogous to
Eq. (16), which when combined with Eqs. (15a) and (24) leads to the differen-
tial equation (17). The application of the inhomogeneous initial condition given by

Eq. (24), and an additional boundary condition specific to the case being considered,
determine the two integration constants γ
1
and γ
2
in the solution
c
n
= γ
1
χ
1
+ γ
2
χ
2
. (25)
2.2 Radial Kinetic Energy: Laguerre Basis
For the case of the radial kinetic energy and a Laguerre basis, the detailed construc-
tion of the J -matrix solutions is given following the general technique out lined in
Section 2.1 The Hamiltonian is
H
0

=−
1
2
d
2
dr

2
+
( + 1)
2r
2
, (26)
while the L
2
expansion set is given by
φ
n
(λr) = (λr)
+1
e
−λr
/
2
L
2+1
n
(λr), n = 0, 1, ,∞, (27)
In essence the infinite matrix problems

φ
m
|

H
0


− k
2

2



˜
S(r)

= 0, m = 0, 1, 2, ,∞, (28a)
J-Matrix Method 25
and

φ
m
|

H
0

− k
2

2



˜
C(r)


= 0, m = 1, 2, ,∞, (28b)
are solved where
˜
S(r) =
∞

n=0
s
n
φ
n
(λr), (13)
˜
C(r) =
∞

n=0
c
n
φ
n
(λr), (19)
subject to the asymptotic boundary conditions
˜
S(r)
∼
r→∞
sin(kr − π


2) (29a)
and
˜
C(r)
∼
r→∞
cos(kr −π

2). (29b)
These solutions have the appropriate asymptotic forms to allow for the formula-
tion of partial wave scattering problems for potentials falling off faster than 1

r
2
at infinity, where the solution of the scattering problem will have the asymptotic
form [3]
⌿
∼
r→∞
sin(kr − π

2) + tan δ cos(kr − π

2), (30)
δ being the scattering phase shift.
From the boundary conditions of Eq. (29),
˜
S(r) is designated the “sine-like”
J-matrix solution, and
˜

C(r) the “cosine-like” J-matrix solution. The sine-like solu-
tion is discussed in Section 2.2.1, where the recurrence relation for the coefficients
{
s
n
}
is solved explicitly, giving closed form expressions. The discussion of
˜
C(r)
is somewhat more complex: In Section 2.2.2, a function
˜
C(r) with the appropriate
cosine-like behavior is constructed such that, for n > 0, the expansion coefficients
{
c
n
}
obey the same recursion scheme as the set
{
s
n
}
; a fact that is an essential in-
gredient of the J -matrix method as will be seen in Sections 3 and 4 and has been
discussed in I and II.
2.2.1 Sine-Like Solution
One of the linearly independent eigenfunctions of the radial kinetic energy,
H
0


=−
1
2
d
2
dr
2
+
( + 1)
2r
2
, (26)
26 H.A. Yamani, L. Fishman
may be taken to be regular at r = 0 and sine-like asymptotically, that is,
⌿
reg
(r)
∼
r→0
r
+1
, (31a)
⌿
reg
(r)
∼
r→∞
sin(kr − π

2) (31b)

where the eigenfunction satisfying Eq. (31) is referred to as the regular solution [3].
A J-matrix solution,
˜
S(r) ≡ ⌿
reg
(r) =
∞

n=0
s
n
φ
n
(λr), (13

)
satisfying the boundary conditions of Eq. (31), is easily found within the context of
the Laguerre set of Eq. (27).
The matrix

φ
n
|

H
0

− k
2


2

|
φ
m

= J
nm
=

∞
0
drφ
n
(λr)

−
1
2
d
2
dr
2
+
( + 1)
2r
2
−
k
2

2

φ
m
(λr),
(32)
may, upon application of the orthogonality and recursion properties of the Laguerre
functions L
2+1
n
(λr) [4], be reduced to the Jacobi (J-matrix) form
J
nm
=−
λη
2
⌫(n + 2 + 2)
⌫(n +1) sin θ

2x(n +  +1)δ
n,m
− nδ
n,m−1
− (n +2 + 2)δ
n,m+1

,
(33)
where
x = cos θ =


η
2
−
1
4

η
2
+
1
4

(34a)
and
η = k

λ =
1
2
cot(θ

2) (34b)
The expansion coefficients
{
s
n
}
which satisfy the matrix equation J · s = 0maybe
determined by the solution of the three-term recursion relation

2x(n + +1)u
n
(x) −(n + 2 + 1)u
n−1
(x) −(n +1)u
n+1
(x) = 0 (35a)
with the initial condition
2x( + 1)u
0
(x) −u
1
(x) = 0, (35b)
where
s
n
(x) =

⌫(n + 1)

⌫(n + 2 + 2)

u
n
(x). (36)
J-Matrix Method 27
Formally, of course, the
{
u
n

}
are given by the Fourier projection [4]
u
n
(x) =

∞
0
dr⌿
reg
(ηr)φ
n
(r)

r (37)
Rather than obtaining the
{
u
n
}
by the direct evaluation of the integral, Eq. (37), a
linear second order differential equation for the
{
u
n
}
is derived. This differential
equation formulation will be utilized for the construction of the cosine-like solution
˜
C(r), where the analog of the projection of Eq. (37) does not exist.

Differentiation of Eq. (37) with respect to x, utilizing the fact that ⌿
reg
is a
function of (ηr), gives, after application of the chain rule, integration by parts, and
application of the Laguerre recursion relations [4],
(x
2
− 1)
du
n
(x)
dx
=
n + 1
2
u
n+1
(x) −
n + 2 + 1
2
u
n−1
(x) (38a)
with the initial condition
(x
2
− 1)
du
0
(x)

dx
=
1
2
u
1
(x). (38b)
Combining Eqs. (35) and (38) gives the differential equation
(1−x
2
)u

n
(x)−xu

n
(x)−

( + 1)
x
2
1 − x
2
− (n + )
2
+( − 1) − 2n − 1

u
n
(x) = 0

(39)
where the differentiation is with respect to x.
Equation (39) is easily solved. Letting u
n
(x) =

1 − x
2

+1
2
v
n
(x)gives
(1 − x
2
)v

n
(x) − (2 + 3)xv

n
(x) − n(n + 2 +2)v
n
(x) = 0 (40)
which is the differential equation satisfied by the Gegenbauer polynomial C
+1
n
(x)
[6]. The general solution of Eq. (39) is then [6]

u
gen
n
(x) = A
n
χ
1
(x) + B
n
χ
2
(x), (41)
where
χ
1
(x) = (sin θ)
+1
C
+1
n
(cos θ), (42a)
χ
2
(x) =
(cos θ

2)
+1
(sin θ


2)

2
F
1
(n +  +
3
2
, −n −  −
1
2
;
1
2
− ;sin
2
θ
2
), (42b)
where again x = cos θ and
2
F
1
(a, b; c; z) is the Gauss hypergeometric function [7].
The coefficients A
n
and B
n
can be determined to within an  dependent factor by
substitution into Eq. (35a), resulting in

28 H.A. Yamani, L. Fishman
u
gen
n
(x) = a

χ
1
(x) +b

χ
2
(x). (43)
The form of u
gen
n
(x) appropriate to the initial condition
2x( + 1)u
0
(x) −u
1
(x) = 0, (35b)
may be determined from Eq. (42) as
u
n
(x) = a

χ
1
(x) (44)

since [4]
2x( +1)C
+1
0
(x) −C
+1
1
(x) = 0 (45)
while χ
2
(x) does not satisfy Eq. (35b) [7]. Substitution of Eq. (44) into Eq. (13)
gives
˜
S(r) = a

(sin θ)
+1
∞

n=0
⌫(n + 1)
⌫(n + 2 + 2)
C
+1
n
(cos θ)φ
n
(λr), (46)
where the requirement that
lim

r→0
k→0

˜
S(kr) −⌿
reg
(kr)

= 0 (47)
determines a

as 2

⌫( + 1) and thus that
s
n
(x) =

2

⌫( + 1)⌫(n + 1)

⌫(n + 2 + 2)

(sin θ )
+1
C
+1
n
(cos θ). (48)

The coefficients
{
s
n
}
of the regular, sine-like, eigenfunction of the radial kinetic
energy have now been determined by the solution of a linear second order differen-
tial equation followed by the imposition of the appropriate boundary conditions.
2.2.2 Cosine-Like Solution
The cosine-like eigenfunction of the radial kinetic energy, which is irregular at the
origin and defined by the conditions
⌿
irreg
(r)
∼
r→0
r
−
, (49a)
⌿
irreg
(r)
∼
r→∞
cos(kr −π

2), (49b)
will be referred to as the irregular solution [3]. For the construction of a cosine-like
J-matrix solution
J-Matrix Method 29

˜
C(r) =
∞

n=0
c
n
φ
n
(λr), (19)
with the asymptotic boundary condition,
˜
C(r)
∼
r→∞
cos(kr −π

2) (50)
it is seen that
˜
C(r) = ⌿
irreg
(r)since
˜
C(r)
∼
r→0
r
+1
as follows from Eq. (19). Thus,

the expansion coefficients
{
c
n
}
cannot be written as a Fourier projection of the form

∞
0
dr⌿
irreg
(ηr)φ
n
(r)

r (51)
in analogy to Eq. (37). A cosine-like J-matrix solution
˜
C(r) must thus be con-
structed with the following requirements: (1)
˜
C(r) should have a cosine-like asymp-
totic form; (2)
˜
C(r) should be regular at the origin; and (3) the coefficients
{
c
n
}
should satisfy the same three-term recursion relation as the set

{
s
n
}
for n > 0. Actu-
ally, from I it is seen that the most general condition in requirement (3) is n ≥ N +1
where N is the number of functions in the subspace onto which the potential V is
projected in the formulation of the model problem. For the purposes of the J-matrix
method, however, it is sufficient to consider the condition n > 0. It is immediately
seen that
˜
C(r) will not satisfy

φ
m
|

H
0

− k
2

2



˜
C(r)


= 0, m = 0, 1, 2, ,∞,as
the cosine-like eigenfunction of H
0

which is linearly independent of ⌿
reg
is ⌿
irreg
which is not regular at the origin.
The function
˜
C(r) satisfying the above conditions is given by the solution of the
equation,

H
0

− k
2

2

˜
C(r) = β
¯
φ
0
(λr) (52)
subject to the boundary conditions
˜

C(r)
∼
r→0
r
+1
, (53a)
˜
C(r)
∼
r→∞
cos(kr −π

2), (53b)
where
¯
φ
0
(λr) = φ
0
(λr)

r⌫(2 + 2) [4]. It is noted that the particular choice of the
inhomogeneity in Eq. (52) gives the infinite matrix problem

φ
m
|

H
0


− k
2

2



˜
C(r)

= βδ
m0
, m = 0, 1, 2, ,∞, (54)
immediately implying that, for n > 0, the
{
c
n
}
and
{
s
n
}
satisfy the same three-term
pure recurrence relation. The parameter β is determined from a Green’s function
construction of the solution to Eq. (52) building in the boundary conditions of
Eq. (53). Using ⌿
reg
and ⌿

irreg
as the two linearly independent solutions of the
30 H.A. Yamani, L. Fishman
homogeneous equation

H
0

− k
2

2

˜
C(r) = 0, the solution of the inhomogeneous
problem of Eq. (52) may be written as [5]
˜
C(r) =−
2β
W

⌿
irreg
(r)

r
0
dr

⌿

reg
(r

)
¯
φ
0
(λr

) + ⌿
reg
(r)

∞
r
dr

⌿
irreg
(r

)
¯
φ
0
(λr

)

,

(55)
where W is the Wronskian and is equal to (−k) [3] for the radial kinetic energy case.
By taking the r →∞limit, β is determined as
β =
−W
2s
0
=
2

k⌫( + 3

2)
√
π(sin θ)
+1
, (56)
where s
0
is the zeroth expansion coefficient of
˜
S(r).
To demonstrate that the
{
c
n
}
satisfy the same differential equation as the
{
s

n
}
the
equation
u
n
(x) =
⌫(n +2 +2)
⌫(n +1)
c
n
(x) =

∞
0
dr
˜
C(r

λ)
φ
n
(r)
r
, (57)
where
˜
C(r) is given by Eq. (55) with β given by Eq. (56), is differentiated with
respect to x. Applying the same procedures that were used in going from Eq. (37)
to (38) yields the differential difference equation of Eq. (38a), which, when com-

bined with the previously derived recursion relation, Eq. (35b), gives the differential
equation (39). The solution u
n
(x) satisfying the initial condition
2x( +1)u
0
(x) −u
1
(x) =−2
+1
⌫

 +
3
2

√
π(sinθ )

(58)
is given by
c
n
(x) = a

χ
1
(x) −
⌫( +
1

/
2
)
√
π
⌫(n + 1)
⌫(n + 2 + 2)
χ
2
(x), (59)
where χ
1
(x)andχ
2
(x) are given by Eq. (42). From the explicit expressions for
˜
S(r)
and
˜
C(r)intermsof⌿
reg
and ⌿
irreg
it follows that
c
n
(−θ) = (−)

c
n

(θ), (60a)
s
n
(−θ) = (−)
+1
s
n
(θ), (60b)
immediately establishing that a

= 0 and giving
c
n
(x) =−
⌫( +
1
/
2
)
√
π
⌫(n +1)
⌫(n +2 +2)
(cos θ

2)
+1
(sin θ

2)


×
2
F
1

n +  +
3
2
, −n −  −
1
2
;
1
2
− ;sin
2
θ
2

. (61)
J-Matrix Method 31
A useful alternative form of c
n
(x)isgivenby[7]
c
n
(x) =−2

⌫( +

1
/
2
)⌫(n + 1)
√
π⌫(n + 2 + 2)
1
(sin θ

2)

×
2
F
1

−n − 2 − 1, n +1;
1
2
− ;sin
2
θ
2

, (62)
which is a finite polynomial in (sin
2
θ

2).

2.3 Radial Kinetic Energy: Oscillator Basis
Within the framework of the oscillator basis
φ
n
(λr) = (λr)
+1
e
−λ
2
r
2

2
L
+1
/
2
n
(λ
2
r
2
), n = 0, 1, ,∞, (63)
the J-matrix defined by
J
nm
=

φ
n

|

H
0

− k
2

2

|
φ
m

=

∞
0
drφ
n
(λr)

−
1
2
d
2
dr
2
+

( + 1)
2r
2
−
k
2
2

φ
m
(λr),
(64)
is a tridiagonal (Jacobi) matrix leading to the fundamental recursion relation
−

2n +  +
3
2
− η
2

u
n

η
2

+

n +  +

1
2

u
n−1

η
2

+(n +1)u
n+1

η
2

= 0, n > 0,
(65)
where again η = k

λ, for the solution of the infinite matrix problems

φ
m
|

H
0

− k
2


2



˜
S(r)

= 0, m = 0, 1, 2, ,∞, (66a)

φ
m
|

H
0

− k
2

2



˜
C(r)

= 0, m = 0, 1, 2, ,∞, (66b)
for the sine-like arid cosine-like J-matrix solutions.
The set

{
u
n
}
satisfies the differential equation
u

n

η
2

+

4n + 2 + 3 − η
2
−
( + 1)
η
2

u
n

η
2

= 0, (67)
where the differentiation is with respect to η. Equation (67) has the general
solution [6]

u
gen
n
(η) = A
n
η
+1
e
−η
2

2
L
+
1
/
2
n

η
2

+B
n
η
−
e
−η
2


2
1
F
1

−n −  −
1
2
,
1
2
− , η
2

,
(68)
32 H.A. Yamani, L. Fishman
where
1
F
1
(a, c, z) is the confluent hypergeometric function [7]. The sine-like solu-
tion is deduced from Eq. (68) by substitution into Eq. (65), imposition of the initial
condition
−

 +
3
2
− η

2

u
0

η
2

+ u
1

η
2

= 0, (69)
and normalization in the manner of Eq. (47), giving
s
n

η
2

=
(−)
n
⌫(n + 1)
⌫(n +  +3

2)
u

n

η
2

=
√
2π
(−)
n
⌫(n + 1)
⌫(n +  + 3

2)
η
+1
e
−η
2

2
L
+
1
/
2
n

η
2


,
(70)
where
˜
S(r) =
∞

n=0
s
n

η
2

φ
n
(λr). (71)
Substitution into Eq. (65), imposition of the initial condition appropriate to the con-
struction of a cosine like solution,
−

 +
3
2
− η
2

u
0


η
2

+ u
1

η
2

=−

2

π⌫

 +
3
2

η
−
e
η
2

2
, (72)
and use of the symmetry conditions given in Eq. (60) give
c

n

η
2

= (−)
n

2
π
⌫

 +
1
2

⌫(n +1)
⌫

n +  +
3
2

η
−
e
−η
2

2

1
F
1

−n −  −
1
2
,
1
2
− , η
2

,
(73)
where
˜
C(r) =
∞

n=0
c
n

η
2

φ
n
(λr). (74)

˜
S(r)and
˜
C(r) are real, regular at the origin, and have sine-like and cosine-like
asymptotic forms, respectively.
2.4 Radial Coulomb Hamiltonian: Laguerre Basis
Within the framework of the Laguerre basis, Coulomb J-matrix solutions
˜
S(r)and
˜
C(r) are constructed, which are regular at the origin and behave asymptotically
as [3]
J-Matrix Method 33
˜
S(r)
∼
r→∞
sin

kr + tln(2kr) −π

2 + σ


, (75a)
˜
C(r)
∼
r→∞
cos


kr +tln(2kr) − π

2 + σ


. (75b)
The Coulomb J matrix
J
nm
=

φ
n
|

H
0
,z
− k
2

2

|
φ
m

=


∞
0
drφ
n
(λr)

−
1
2
d
2
dr
2
+
( + 1)
2r
2
+
z
r
−
k
2
2

φ
m
(λr)
(76)
is a tridiagonal form, yielding the recurrence relation


(n +  + 1)

x
2
+ 1
x

−it

x
2
− 1
x

v
n
(x)
−(n + 1)v
n+1
(x) −(n +2 + 1)v
n−1
(x) = 0
, n > 0 (77)
where
x = e
iθ
=−

1

2
−iη

1
2
+iη

(78)
and
s
n
(x) =
2

⌫(n + 1)
|
⌫( + 1 −it)
|
⌫(2 + 2)⌫(n +2 + 2)
e
πt
/
2
v
n
(x) (79)
for the solution of the infinite matrix equations

φ
m




H
0
,z
− k
2

2



˜
S(r)

= 0, m = 0, 1, 2, ,∞, (80a)

φ
m



H
0
,z
− k
2

2




˜
C(r)

= 0, m = 0, 1, 2, ,∞, (80b)
for the sine-like and cosine-like J-matrix solutions.
The
{
v
n
}
satisfy the differential equation
(x
2
− 1)v

n
(x) +
x
2
− 1
x
v

n
(x) −



x
2
− 1
x
2


(n + )
2
− ( − 1) + 2n +1 −t
2

+

x
2
+ 1
x
2

x
2
+ 1
x
2
− 1

( + 1) − 2it(n + + 1)



v
n
(x) = 0
(81)
where the differentiation is with respect to x = exp(iθ)andwheret =−z

k is
considered to be independent of x. The derivation of this equation is discussed in
the Appendix. The general solution of Eq. (81) is [7]
34 H.A. Yamani, L. Fishman
v
gen
n
(x) = A
nt
(sin θ)
+1
e
θt
e
−inθ
2
F
1

−n,+ 1 −it;2 + 2; 1 −e
2iθ

+ B
nt

(sin θ)
it
e
−i(n++1)θ
2
F
1

−  −it,+ 1 − it; n +  + 2 −it;
×

1 − e
2iθ

−1

(82)
The sine-like solution is determined from Eq. (82) by substitution into Eq. (77),
imposition of the initial condition

( + 1)

x
2
+ 1
x

− it

x

2
− 1
x

v
0
(x) −v
1
(x) = 0, (83)
and by application of a normalization procedure discussed in the Appendix, giving
s
n
(θ) = 2

n!
|
⌫( + 1 −it)
|
⌫(n + 2 + 2)
(sin θ)
+1
e
θt
exp

ε +
1
2

πt


P
+1
n

cos θ ;
2z
λ
; −
2z
λ

,
(84)
where
ε =

−1forθ ∈ [0, +π ]
+1forθ ∈ [0, −π ]
(85)
and P
λ
n
(z; a; b) is the Pollaczek polynomial [4] as discussed in the Appendix. The
cosine-like solution is determined by substitution into Eq. (77), imposition of the
initial condition

( + 1)

x

2
+ 1
x

−it

x
2
− 1
x

v
0
(x) −v
1
(x) =
−2
−2
e
−θt
[⌫(2 + 2)]
2
(sin θ)

|
⌫( + 1 −it)
|
2
(86)
and application of a limiting procedure as discussed in the Appendix, giving

c
n
+is
n
=
−n!e
iσ

e
πt
/
2
e
−θt
e
−i(n+1)θ
(2 sin θ )

⌫(n +  +2 −it)
2
F
1
(−−it, n+1; n++2−it; e
−2iθ
). (87)
The functions
˜
S(r)and
˜
C(r) are real and reduce to the radial kinetic energy results

when z = 0.
3 Potential Scattering
In this section, it will be assumed that the potential V does not couple angular mo-
mentum eigenstates; the generalization to the case where coupling occurs is straight-
forward and will be considered in the multi-channel case discussed in Section 4.
Thus, the good angular momentum quantum number  will be suppressed, it being
implicitly assumed that a definite partial wave is under consideration. The aim of
J-Matrix Method 35
this section is then to determine the phase shift caused by the potential V with
respect to the uncoupled Hamiltonian H
0
which may be taken to be the th partial
wave kinetic energy or Coulomb Hamiltonian.
As alluded to in the Introduction, and motivated in I, the potential V is approxi-
mated by truncating its representation in the function space
{
φ
n
}
to a finite, N × N,
representation V
N
defined by
V
N
nm
=

∞
0

φ
n
(λr)V (r)φ
m
(λr)dr, n, m ≤ N − 1,
0, otherwise.
(88)
The problem is then to solve

⌽
m
|

H
0
+ V
N
− k
2

2

|
⌿
E

= 0, m = 0, 1, 2, ,∞, (89)
where
⌿
E

=
∞

n=0
d
n
φ
n
(λr). (90)
The form of V
N
as defined in Eq. (88) is such, however, that it only couples the first
N functions φ
m
, m = 0, 1, 2, ,N −1, in the infinite function space. Thus, outside
the space spanned by these N basis functions the generalized sine-like and cosine-
like solutions associated with the generalized H
0
problem discussed in Section 2 are
valid. This leads to the following form for the wavefunction ⌿
E
(r),
⌿
E
(r) = ⌽(r) + S(r) +tC(r), (91)
where
⌽(r) =
N−1

n=0

a
n
φ
n
(λr), (92a)
S(r) =
∞

n=N
s
n
φ
n
(λr), (92b)
C(r) =
∞

n=N
c
n
φ
n
(λr), (92c)
and t is the tangent of the phase shift caused by V
N
with respect to H
0
. Note that
the first N terms in the expansions of
˜

S(r)and
˜
C(r) have been incorporated into
the ⌽(r) term, the remainder of the expansions being designated as S(r)andC(r),
respectively. The sets of coefficients
{
s
n
}
and
{
c
n
}
are just those that were determined
in Section 2 and are dependent upon the particular H
0
and basis set
{
φ
n
}
being
considered. The solution (89) given in Eq. (91) contains N + 1 unknowns, (t,
{
a
n
}
,
n = 0, 1, ,N −1).

36 H.A. Yamani, L. Fishman
By returning to Eq. (89), it is immediately established that it is satisfied for m ≥
N + 1: Since V
N
is de fined to be zero in this region of the function space and due
to the tridiagonal representation of

H
0
− k
2

2

,

φ
m
|

H
0
+ V
N
− k
2

2

|

⌿
E

=

φ
m
|

H
0
− k
2

2

|
S + tC

(93a)
=

φ
m
|
J
|
S + tC

(93b)

=
∞

n=0
J
mn
(s
n
+ tc
n
) (93c)
= J
m,m−1
(s
m−1
+ tc
m−1
) + J
m,m
(s
m
+ tc
m
) + J
m,m+1
(s
m+1
+ tc
m+1
) (93d)

= 0. (93e)
Equation (93e) follows from the recursion relation satisfied by both s
n
and c
n
.The
remaining N +1 conditions corresponding to the cases m = 0, 1, , N are sufficient
to determine the N + 1 unknowns, and by following the analysis presented in I the
resulting system matrix is obtained:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
(J + V )
0,0
··· (J + V )
0,N−2
(J + V )
0,N−1
| 0

(J + V )
1,0
··· (J + V )
1,N−2
(J + V )
1,N−1
| 0
···|·
···|·
···|·
(J + V )
N−2,0
···(J + V )
N−2,N−2
(J + V )
N−2,N−1
| 0
(J + V )
N−1,0
···(J + V )
N−1,N−2
(J + V )
N−1,N−1
| J
N−1,N
c
N
−−−− −−− −−−−− −−−−− | −−−−
0 ··· 0 J
N,N−1

|−J
N,N−1
c
N−1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
×
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢

⎢
⎢
⎣
a
0
a
1
·
·
·
a
N−2
a
N−1
−
t
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
0
·
·
·
0
−J
N−1,N
s
N
−−−−
J
N,N−1
s
N−1

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(94)
Equation (94) can be immediately solved for t by standard techniques. In partic-
ular, an expression for t = tan δ can be obtained by pre-diagonalizing the inner
N × N matrix

H
0
+ V
N
− k
2

2

nm
with the energy independent transformation ⌫

defined by

˜
⌫

H
0
+ V
N
− k
2

2

⌫

nm
= (E
n
− E)δ
nm
, (95)
J-Matrix Method 37
where
{
E
n
}
, are the Harris eigenvalues. Augmenting ⌫ to be the (N +1) ×(N +1)
matrix,

⌫
A
=

⌫ 0
01

, (96)
and applying it to Eq. (94) gives
t = tan δ =−
s
N−1
+ ν(E)J
N,N−1
s
N
c
N−1
+ ν(E)J
N,N−1
c
N
, (97)
where
ν(E) =
N−1

m=0
⌫
2

N−1,m
E
m
− E
. (98)
At the positive Harris eigenvalues E
n
[8], tan δ reduces to
tan δ(E
n
) = s
N
(η
n
)

c
N
(η
n
),η
n
= k
n

λ. (99)
Also, at the positive energies E
μ
,whereν(E
μ

) = 0, tan δ becomes
tan δ(E
μ
) =−s
N−1
(η
μ
)

c
N−1
(η
μ
),η
μ
= k
μ

λ. (100)
Equations (94), (97), (99), and (100) are the generalizations of the corresponding
results obtained in I.
4 Multi-Channel Scattering
The potential scattering results of the previous section will now be generalized
to include collisions with targets having internal states. Due to the generaliza-
tions developed for solving the H
0
problem, the results of which are given in
Sections 2.2–2.4, collisions with charged, as well as neutral targets, can be consid-
ered, while employing the appropriate basis set (Laguerre or oscillator in the neutral
case, and Laguerre in the charged case) to describe the projectile wavefunction.

Since the method is formulated in terms of the close-coup1ing [9] equations, ex-
change can be treated by the inclusion of the appropriate nonlocal potential followed
by its truncation in the J-matrix sense as was done in II.
The target will be described by the collective coordinate ρ, and its dynamics by
H
t
(ρ). It is assumed that the target possesses a discrete set of L
2
eigenfunctions
R
γ
(ρ) and, further, that the collective quantum number γ includes the total orbital
quantum number of the target, L
t
, its projection on some specified direction (the
z axis), M
t
, in addition to the quantum numbers μ that are needed further to com-
pletely define the target states. If the target has a dense or continuous spectrum, the
method of pseudotarget states may be employed [2, 10]. It then follows that
38 H.A. Yamani, L. Fishman
H
t
(ρ)R
γ
(ρ) = E
μ,L
t
R
γ

(ρ), (101)
where γ =
{
μ, L
t
, M
t
}
.
The wavefunction ⌰ describing the projectile-target system satisfies the follow-
ing Schr¨odinger equation:
[H
t
(ρ) + H
0
(r) + V (r,ρ) − E] ⌰(r,ρ) = 0, (102)
where r is the projectile coordinate, H
0
(r) =−
1
2
∇
2
r
+z
/
r, its Hamiltonian in atomic
units, and V(r,ρ), the interaction between the projectile and the target. In the neutral
case z is equal to zero.
For most cases of interest, such as the scattering of electrons by light atoms, the

total angular momentum of the system, L, its projection along the z axis, M,as
well as the parity, ⌸, are conserved in the collision process. Therefore, it is more
convenient to use a representation which is diagonal in these quantum numbers.
Coupling the projectile with the target [9] in a picture with definite total L and total
M leads to defining
χ
⌫
(ˆr,ρ) =

m,M
t
C(L
t
L, mM
t
M)Y
m
(ˆr)R
γ
(ρ), (103)
where C(L
t
L, mM
t
M) is the Clebsch-Gordan coefficient, ⌫, the channel index
{
μL
t
LM
}

,andY
m
(ˆr) the spherical harmonic functions. It is noted that χ
⌫
is an
eigenfunction of L and M, satisfying the same equation as R
γ
, namely,
H
t
(ρ)χ
⌫
(ˆr,ρ) = E
μ,L
t
χ
⌫
(ˆr,ρ), (104)
where L satisfies the triangular relations,
|
L
t
− 
|
≤ L ≤ L
t
+. Since the function
χ
⌫
is composed of L

2
functions and is an eigenfunction of H
t
(ρ), the set
{
χ
⌫
}
may
be taken to be orthonormal,

χ
⌫
|
χ
⌫


= δ
⌫⌫

= δ
μμ

δ
L
t
L

t

δ


, (105)
where ⌫ =
{
μL
t
LM
}
and ⌫

=

μ

L

t


LM

.
Within a definite LM picture, the Hilbert space of the system is spanned by the set

|
χ
⌫




φ
(,⌫)
n

(106)
for all μ, L
t
, ,andforn = 0, 1, ,∞. For neutral targets the set

φ
(,⌫)
n

can be
either the Laguerre functions
φ
(,⌫)
n
(λ
⌫
r) = (λ
⌫
r)
+1
e
−λ
⌫
r

/
2
L
2+1
n
(λ
⌫
r), n = 0, 1, ,∞, (107)
J-Matrix Method 39
or the oscillator functions
φ
(,⌫)
n
(λ
⌫
r) = (λ
⌫
r)
+1
e
−λ
2
⌫
r
2

2
L
+1
/

2
n
(λ
2
⌫
r
2
), n = 0, 1, ,∞, (108)
while for charged targets the set

φ
(,⌫)
n

can only be the Laguerre functions. Note
that the projectile basis can be made channel dependent through the channel depen-
dent scaling parameter λ
⌫
.
In general it is not possible to solve Eq. (102) for ⌰(r,ρ) exactly in the Hilbert
space of Eq. (106). By following the procedure in I, the interaction potential V is
approximated by
˜
V , which is defined by the following truncation scheme:
˜
V
⌫⌫

nn


=

χ
⌫
φ
(,⌫)
n



V (r,ρ)



χ
⌫

φ
(

,⌫

)
n


for ⌫, ⌫

≤ N
c

, n ≤ N
⌫
− 1, n

≤ N
⌫

− 1
0, otherwise
(109)
where N
⌫
is the truncation limit in the channel ⌫ and N
c
is the total number of
channels that are allowed to couple through the potential.
It is now proposed to solve the model equation

H
t
(ρ) + H
0
(r) +
˜
V (r,ρ) − E

⌰(r,ρ) = 0 (110)
exactly in the Hilbert space of Eq. (106). There will be as many independent solu-
tions ⌰
⌫

as there are open channels. By following the approach taken in I, ⌰
⌫
is
expanded as
⌰
⌫
(r,ρ) = ⌽
⌫
(r,ρ) +
χ
⌫
(ˆr,ρ)S
⌫
(r)
√
k
⌫
+
N
c

⌫

χ
⌫

(ˆr,ρ)C
⌫

(r)R

⌫

⌫
√
k
⌫

,
⌫ = 1, 2, ,N
0
(111)
where N
0
is the number of open channels. The scattering matrix S
LM
is related to
the reactance matrix elements by the relation,
S
LM
= e
iσ
(1 + i[R]) (1 −i[R])
−1
e
iσ
, (112a)
where [R]istheN
0
× N
0

open-channel part of R and

e
iσ

⌫⌫

= e
iσ

δ
⌫⌫

. (112b)
In Eq. (111), the quantity k
⌫
is the wave number of the scattered electron and is
given by
k
⌫
=

2


E − E
μ,L
t



. (113)
40 H.A. Yamani, L. Fishman
Furthermore, for an open channel ⌫,
S
⌫
(r) =
∞

n=N
⌫
s
n
(k
⌫
)φ
(,⌫)
n
(λ
⌫
r), (114a)
C
⌫
(r) =
∞

n=N
⌫
c
n
(k

⌫
)φ
(,⌫)
n
(λ
⌫
r), (114b)
where
{
s
n
}
and
{
c
n
}
are given by the results of Section 2, while, for a closed channel
⌫, the cosine-like term C
⌫
(r) is replaced by the linear combination
C
⌫
(r) +iS
⌫
(r) (115)
and evaluated at k
⌫
= i
|

k
⌫
|
. The resulting function correctly describes the exponen-
tially decaying closed channel asymptotic behavior. In the neutral case, the function
of Eq. (115) is related to the spherical Hankel function of the first kind, h
(1)

(z), eval-
uated at z = i
|
k
⌫
|
r. The internal function, ⌽
⌫
(r,ρ), which describes the scattering
process at close distances, is written as
⌽
⌫
(r,ρ) =
N
c

⌫

N
⌫

−1


n=0
a
⌫

⌫
n
χ
⌫

(ˆr,ρ)φ
(

,⌫

)
n
(λ
⌫

r). (116)
The remainder of this section briefly demonstrates that the wavefunction ⌰ is ca-
pable of being an exact solution of the model Hamiltonian in the Hilbert space. This
is accomplished by uniquely determining the unknowns

a
⌫

⌫
n

, R
⌫

⌫

. It is required
that all projections by

χ
⌫

φ
(

,⌫

)
n




from the left-hand side of Eq. (110) vanish:

χ
⌫

φ
(


,⌫

)
n





H
t
(ρ) + H
0
(r) +
˜
V (r,ρ) − E




⌰(r,ρ)

= 0. (117)
For the case of ⌫ = 1, 2, ,N
0
, ⌫

= 1, 2, ,N
c
,andn


= 0, 1, ,N
⌫

,
N
0


N
c
⌫

=1
(N
⌫

+ 1)

equations are obtained which is equal to the total number of
unknowns: N
0


N
c
⌫

=1
N

⌫


of the a’s and N
0
N
c
of the R’s. It only remains now
to show that for all other cases Eq. (117) is satisfied. The assertion is clear when
⌫

> N
c
. Suppose now that ⌫

≤ N
c
,butthann

≥ N
⌫

+ 1. Then
˜
V
⌫

⌫

n


n

= 0by
definition and Eq. (117) reduces to

φ
(

,⌫

)
n







−
1
2
d
2
dr
2
+



(

+ 1)
2r
2
+
z
r
−

E − E
μ

,L

t


×





N
⌫

−1

n=0

a
⌫

⌫
n
φ
(

,⌫

)
n
+
S
⌫

√
k
⌫

δ
⌫⌫

+
C
⌫

R
⌫


⌫
√
k
⌫


= 0 (118)
J-Matrix Method 41
where z = 0 in the neutral target case. Since the operator appearing in Eq. (118) is
tridiagonal in the

φ
(

,⌫

)
n

representation, the contribution of the S and C functions
vanish since their expansion coefficients
{
s
n
}
and
{
c
n
}

satisfy the resulting three-
term recursion relation. Since n

≥ N
⌫

+1 while n ≤ N
⌫

−1, the contribution of
the sum term in Eq. (118) vanishes, proving that Eq. (118) is identically true.
As in I, the nontrivial equations can be arranged such that the L
2
matrix elements
of

H
t
+ H
0
+
˜
V

appear in an inner block. Additionally, one extra row and column
are added to this block for each channel ⌫ ≤ N
c
. The right-hand side driving term
and the solution vector, containing the a
⌫⌫


n
and R
⌫⌫

, have as many columns as open
channels. The R matrix can then be extracted by solving the resulting linear equa-
tions by standard techniques. After having done so, the S
LM
matrix is constructed
via Eq. (112), from which physical quantities are then obtained: e.g., the cross sec-
tion for the transition

μ
0
L
0
t
→ μL
t

, averaged over M
0
t
states and summed over
M
t
states, is given by
σ


μ
0
L
0
t
→ μL
t

=
π

2L
0
t
+ 1

k
2
⌫
0

L
0
L
(2L + 1)


δ
⌫
0

⌫
− S
LM
⌫
0
⌫


2
. (119)
Other physical quantities of interest can be similarly constructed [9].
5 Discussion
The discussion in I compared the J-matrix approach with the R-matrix, separable
kernel, and Fredholm methods. The remarks were of such a general nature as to
apply to the extensions made in this chapter. In particular, there are two points that
should be stressed. First, no Kohn-type pseudo-resonances are expected to appear in
the computed cross sections. This can be demonstrated by bounding these quantities
for all energies [11]. Secondly, since H
0
is solved analytically within the function
space, it is expected that small basis sets would be adequate to account for the ad-
dition of the approximate potential. The resulting physical quantities contain first
order errors; however, the analytic nature of the solutions allows for variational
correction. This can be accomplished through the application of the Kato correc-
tion [12] as discussed in I, and results in reducing the errors to second order.
Presently, work is being done on some of the more mathematical aspects of the
J- matrix method. These include the sense of convergence of the expansions for the
sine-like and cosine-like functions, possible analytic approaches to the second order
Kato correction, and the generality of the solution scheme for the H
0

problem. In
addition, the J-matrix method is being applied to the e—He
+
scattering calculation.
Acknowledgments The authors are grateful to Professor William P. Reinhardt for encourage-
ment and support of this work along with a critical reading of the manuscript. Additional thanks
are expressed to Dr. E. J. Heller for continued interest in this work and a critical reading of the
manuscript. Helpful conversations with J. Broad, T. Murtaugh, and J. Winick are acknowledged.
This work was supported by grants from the National Science Foundation and the Camille and
Henry Dreyfus Foundations.