solving a three-body continuum coulomb problem with quasi ... · j. at. mol. sci. doi:...

J. At. Mol. Sci.doi: 10.4208/jams.121312.012013a

Vol. 4, No. 4 , pp. 302-320November 2013

Solving a three-body continuum Coulomb problem

with quasi-Sturmian functions

S. A. Zaytseva,∗and G. Gasaneob

a Pacific National University, Khabarovsk, 680035, Russiab Departamento de Fısica, Universidad Nacional del Sur, 8000 Bahıa Blanca, BuenosAires, Argentina; Consejo Nacional de Investigaciones Cientıficas y Tecnicas,Argentina

Received 13 December 2012; Accepted (in revised version) 20 January 2013Published Online 30 August 2013

Abstract. The scattering problem of three particles interacting via Coulomb poten-tials is studied using generalized parabolic coordinates. The scattering solutions areobtained by solving a driven equation. The ‘perturbation’ operator appearing in thedriven term is the non-orthogonal part of the kinetic energy operator. The approxi-mated solution appearing in the driven term is the product of two two-body Coulombwave functions. As a test for our proposal, a simple two-dimensional model problemhas been solved numerically by using so called parabolic quasi-Sturmian basis repre-sentation. Convergence of the solution has been obtained as the basis set is enlarged.

PACS: 34.50.Fa, 34.10.+x, 03.65.Nk

Key words: three-body coulomb system, parabolic coordinates, driven equation, quasi-Sturmians,convergence

1 Introduction

The three-body continuum Coulomb problem is one of the fundamental unresolved prob-lems of theoretical physics. In atomic physics, a prototype example is a two-electron con-tinuum which arises as a final state in electron-impact ionization and double photoion-ization of atomic systems. Several discrete-basis-set methods for the calculation of suchprocesses have recently been developed including convergent close coupling (CCC) [1,2],the Coulomb-Sturmian separable expansion method [3,4], the J-matrix method [5–7], theGeneralized Sturmian approach [8, 9]. In all these approaches the continuous Hamilto-nian spectrum is represented in the context of complete square integrable bases. Despitethe enormous progress made so far in discretization and subsequent numerical solutions

∗Corresponding author. Email address: [email protected] (S. A. Zaytsev)

http://www.global-sci.org/jams 302 c©2013 Global-Science Press

S. A. Zaytsev and G. Gasaneo / J. At. Mol. Sci. 4 (2013 ) 302-320 303

of three-body differential and integral equations of the Coulomb scattering theory, a num-ber of related mathematical problems remain open. Actually, the use of a product of twofixed charge Coulomb waves for the two outgoing electrons as an approximation to thethree-body continuum state is typical of these approaches. As a consequence, a long-range potential appears in the kernel of the corresponding Lippmann-Schwinger (LS)equation. Since this integral equation is, in principle, non-compact, its formal solutiontherefore should be divergent. Note, however that for the two-body problem this typeof definition for the ‘free particle solution’ is not leading to a divergent solution [10]. Inaddition, for the three-body case approaches such as the exterior complex scaling [11]and the generalized Sturmian approaches [8, 9] lead to correct solutions for the drivenequation from which the LS equations are derived. One of the aims of this paper is to un-derstand the reason for that differences between the solution corresponding to LS typeand driven equations.

On the other hand, it is well known [12, 13] that for large particle separations (in theΩ0 region) the Schrodinger equation for a three-body Coulomb system is separable interms of so called generalized parabolic coordinates

ξ j,ηj

, j=1,2,3 [13, 14]. Moreover,a representation of the corresponding Green’s function operator has been derived in [15].Thus, at first glance it would seem that a formal solution for the three-body Coulombproblem can be expressed in the form of a Lippmann-Schwinger-type equation, wherethe potential operator, which coincides with the non-orthogonal part of the kinetic en-ergy operator, is expressed in terms of second partial mixed derivatives whit respect tothe parabolic coordinates. No complete studies of the compactness of the kernel of thisintegral equation can be found in the literature [16]. Actually, a differential operator ofthis type seems to be unbounded in a Hilbert space and therefore the corresponding LSequation could presents difficulties in its formal solution. To avoid these problems, analternative approach can be performed by considering an inhomogeneous Schrodingerequation whose driven term is square integrable. In this paper we formulate a procedurefor solving the driven equation using so called quasi-Sturmian (QS) functions. UnlikeSturmian functions (see, e. g., [17–20] and references therein), which are eigensolutionsof a Sturm-Liouville differential or integral equation, and form a complete set of basisfunctions, the QS are constructed from square-integrable basis functions with the helpof an appropriate Coulomb Green’s function operator. In order to test practically the QSapproach and the solution to an equation of a driven type instead of LS equations we con-sider a simple two-dimensional model problem on the plane (ξ1,ξ3). Here the total waveoperator, aside from the one-dimension Coulomb wave operators h1 and h3, contains the

‘perturbation’ term ∂2

∂ξ1 ∂ξ3.

This paper is organized as follows. In Section II we introduce the notations, recallthe generalized parabolic coordinates definition and express a formal solution for thethree-body Coulomb problem in the form of a driven equation. In Section III we presentthe quasi-Sturmian functions and its properties. We also present its representation interms of Laguerre basis functions and some of its properties. In Section IV a simple two-dimensional model is presented and used to test the parabolic QS approach. The calcu-

304 S. A. Zaytsev and G. Gasaneo / J. At. Mol. Sci. 4 (2013 ) 302-320

lations of the continuum wave function is performed. A study of the convergence rate asa function of the number of the quasi-Sturmians used is presented. Some conclusion aregiven in Section V. Atomic units are used throughout.

2 Coulomb three-body system in parabolic coordinates

2.1 General considerations

We consider three particles of masses m1, m2, m3, charges Z1, Z2, Z3 and momenta k1, k2,k3. The Hamiltonian of the system in the center of mass frame of reference is given by

H=− 1

2µ12∆R−

1

2µ3∆r+

Z1Z2

r12+

Z2Z3

r23+

Z1Z3

r13, (1)

where rls denotes the relative coordinates

rls = rl−rs, rls = |rls|, (2)

R and r are the Jacobi coordinates

R= r1−r2, r= r3−m1r1+m2r2

m1+m2. (3)

The reduced masses are defined as

µ12=m1m2

m1+m2, µ3=

m3(m1+m2)

m1+m2+m3. (4)

In the Schrodinger equation

HΦ=EΦ (5)

the eigenenergy E>0 is given by

E=1

2µ12K2+

1

2µ3k2, (6)

where K and k are the momenta conjugate to the variables R and r. By substituting

Φ= ei(K·R+k·r)Ψ (7)

into (5), we arrive at the equation for the reduced wave function Ψ

[

− 1

2µ12∆R−

1

2µ3∆r−

i

µ12K·∇R−

i

µ3k·∇r+

Z1Z2

r12+

Z2Z3

r23+

Z1Z3

r13

]

Ψ=0. (8)


The leading-order asymptotic terms of Ψ in the Ω0 domain are expressed in terms of thegeneralized parabolic coordinates [13]

ξ1= r23+k23 ·r23, η1= r23−k23 ·r23,

ξ2= r13+k13 ·r13, η2= r13−k13 ·r13,

ξ3= r12+k12 ·r12, η3= r12−k12 ·r12,

(9)

where kls =klms−ksml

ml+msis the relative momentum, kls =

klskls

and kls = |kls|. The operator in

the square brackets, denoted by D, can be decomposed into two terms [13]

D= D0+D1, (10)

where the operator D0 contains the leading term of the kinetic energy and the total po-tential energy

D0=3

∑j=1

1µls(ξ j+ηj)

[

hξ j+ hηj

+2klstls

]

,

for j 6= l, s and l< s,

(11)

hξ j=−2

(

∂

∂ξ jξ j

∂

∂ξ j+iklsξ j

∂

∂ξ j

)

, (12)

hηj=−2

(

∂

∂ηjηj

∂

∂ηj−iklsηj

∂

∂ηj

)

. (13)

Here tls =Zl Zsµls

klsand µls =

mlms

ml+ms. D1 represents the remaining part of the kinetic energy

operator [13] which, in the case of the (e−, e−,He++)= (123) system with m3 =∞, takesthe form [21]

D1=2

∑j=1

(−1)j+1[

u−j ·u−

3∂2

∂ξ j ∂ξ3+u−

j ·u+3

∂2

∂ξ j ∂η3

+ u+j ·u−

3∂2

∂ηj ∂ξ3+u+

j ·u+3

∂2

∂ηj ∂η3

]

,

(14)

whereu±

j = rls∓kls. (15)

The asymptotic behavior of Ψ is determined by the operator D0. In particular, thereexist solutions to the equation

D0ΨC3=0, (16)

which satisfy the Redmond conditions in Ω0. These solutions are the well-known C3wave function. ΨC3 is expressed in terms of a product of three Coulomb waves. Forexample, ΨC3 with pure outgoing behavior is written as

ΨC3 =3

∏j=1

1F1

(

itls, 1;−iklsξ j

)

. (17)

In turn, D1 is regarded as a perturbation which does not violate the asymptotic conditions[13, 14].


2.2 Formal solution of the problem

At first sight, given the Green’s function operator G= D−10 (see [15]), one could take into

account the non-orthogonal term D1 of the kinetic energy operator by putting it into thekernel of the Lippmann-Schwinger type equation

Ψ=ΨC3−GVΨ,

V ≡ D1.(18)

If the kernel GV is compact, then the integral equation (18) can be solved by an algebraicmethod based on the fact that a compact operator may be uniformly approximated byoperators of finite rank. For this purpose a set of square-integrable parabolic Laguerrebasis functions [22]

|N〉≡BN(ξ,η)=3

∏j=1

φnj m j

(

ξ j,ηj

)

, (19)

φnj m j

(

ξ j,ηj

)

=ψnj

(

ξ j

)

ψm j

(

ηj

)

, (20)

ψn (x)=√

2bj e−bjxLn(2bjx), (21)

could be used. The index N lists all the indexes of the basis function N=n1,m1,n2,m2,n3 ,m3 and the argument (ξ,η) of the function BN(ξ,η) represents in compact form the de-pendence on all the parabolic coordinates. The basis functions (20), (21) are parametrizedwith different Coulomb Sturmian exponents bj for each pair

ξ j,ηj

, j = 1,3. Thus, the

operator V is represented by its projection VN onto a subspace of basis functions,

VN =N0

∑N,N′=0

|N〉〈N|V∣

∣N′⟩⟨N′∣∣, (22)

and the solution Ψ of the problem is obtained for VN . Inserting VN into Eq. (18) thenleads to a finite matrix equation for the expansion coefficients [a]N= 〈N|Ψ 〉,

a= a(0)−GV a, (23)

which has the solution

a=(1+GV)−1 a(0). (24)

Here [G]NN′ = 〈N|G |N′〉 and [V ]NN′ = 〈N|V |N′〉 are the Green’s function operator and

potential operator matrices of the size N0+1, and a(0) is the coefficient vector of ΨC3. Thewave function Ψ is expressed in terms of the solution of Eq. (23)

Ψ=ΨC3−N0

∑N=0

[C]N G |N〉, (25)


where C=V a.We performed different studies of this equation and we found that the kernel is not

compact when is expressed in terms of L2-spaces. Actually the problem is that the stan-dard L2 bases do not possess the appropriated asymptotic behavior of the problem. Thus,this means that the asymptotic behavior has to be constructed and then the perturbationoperator D1 (14) seems to be not bounded. However, if the basis already possesses theasymptotic behavior of the problem, then, the operator turns to be of a short range andthen becomes compact and manageable.

Even when what we just mentioned is possible, we will explore an alternative ap-proach to the problem based on the study of the driven equation

[

D0+D1

]

Ψsc=−D1ΨC3, (26)

where the wave function Ψ is split into an outgoing (incoming) part ΨC3 and a scatteringΨsc part

Ψ=Ψsc+ΨC3. (27)

Note that the inhomogeneity in (26) is a square-integrable function. Eq. (25) in turn givesa hint as to how the solution Ψsc to (26) may be constructed with the help of the square-integrable basis (19). Namely, we suppose that the wave function Ψ can be expressed inthe form (25), i. e., propose to expand Ψsc as

Ψsc= ∑N=0

[c]N |QN〉 , (28)

where|QN〉≡G |N〉. (29)

We will call |QN〉 quasi-Sturmian functions. The word ‘quasi’ refers to the fact that inorder to obtain these functions there is no need to solve a Sturm-Liouville equation.

3 Quasi Sturmian functions

According with the definition of Eq. (29) the QS functions satisfy the following drivenequation

D0QN(ξ,η)=BN(ξ,η) (30)

and they possess the same asymptotic behavior as the Green’s function G. In this casewe are using on the right-hand-side the Laguerre type basis functions BN(ξ,η), but anybasis set can be used. However, to preserve the asymptotic behavior of the QN functions,the extent of the basis functions in the configuration space has to be finite. A closed formexpression for the Green’s function G was given in Ref. [23] and this allows one to expressthe QS functions in terms of special functions. The right hand side of (30) depends on theindexes N, thus for each set of values N we have different functions QN. All of thesefunctions form a complete basis set even when they are not orthogonal.


When solving Eq. (26) with the proposal (28) we enforce the solution to possess thecorrect outgoing behavior of the scattering function. This is similar to what it is observedwhen generalized Sturmian functions are used [8, 9]. The completeness of the QN, theshort-range of both the right-hand-side and D1QN allow to assure that this expansionconverges. To exemplify this affirmation we will solve in the following sections a two-dimensional model problem presented in [21]. We will use a product of one-dimensionalQS functions obtained from one-dimensional Green’s function

Qn(k,ξ)≡∫

dξ′G(+)(k; ξ,ξ′)ψn(ξ′). (31)

This will allow us to probe the convergence of the expansion of two-dimensional scatter-ing wave function before considering a very elaborate and cumbersome six-dimensionalcase as required for the full three-body problem.

3.1 Definitions and asymptotic behaviors

We start with the Green’s function which satisfies the equation

[

−2∂

∂ξξ

∂

∂ξ−2ikξ

∂

∂ξ+2kt

]

g(+)(

k; ξ,ξ′)

=δ(ξ−ξ′) (32)

One of the solutions of the equation (32) can be written

gs(+)(

k; ξ,ξ′)

=Γ(it)

21F1(it, 1,−ikξ<)U(it, 1;−ikξ>) eikξ ′ . (33)

Lets us define the functions

qsn(k,ξ)≡

∫

dξ′gs(+)(k; ξ,ξ′)ψn(ξ′), (34)

It follows from (33) and (34) that

qsn(k,ξ) ∼

ξ→∞vn U(it, 1;−ikξ) , (35)

where

vn ≡Γ(it)

2

∞∫

0

dξ1F1(it, 1,−ikξ) eikξ ψn(ξ)

=1√2b

(

1+ω

2ω

)1−it

(−ω)−n Γ(it)Γ(n+1−it)

n!Γ(1−it)2F1(−n, it; it−n; ω),

ω=b−ik

b+ik.

(36)


0 10 20 30 40 50

-1

0

1Z = -2; k = 1; b = 1; n = 0

n = 1 n = 5 n = 10 n = 15 n = 20

QnRe

Figure 1: Real parts of QS functions Qn (51), (43), (40) for Z=−2, k=1, b=1, µ= 12 .

0 10 20 30 40 50

-1

0

1Z = -2; k = 1; b = 1;

n = 0 n = 1 n = 5 n = 10 n = 15 n = 20

QnIm

Figure 2: The same as in Fig. 1 but for the imaginary parts.

Using the integral representation for the Whittaker functions [24] we can write

gs(+)(k; ξ,ξ′)=1

2ei k

2 (ξ′−ξ)

∞∫

0

dzexp

ik

2(ξ+ξ′)cosh(z)

×[

coth(z

2)]−2it

I0

(

−ik√

ξξ′ sinh(z))

,

(37)


0 10 20 30 40 50

-2

-1

0

1

2

Z = 1; k = 0.1; b = 0.1;

n = 0 n = 1 n = 5 n = 10 n = 15 n = 20

QnRe

Figure 3: The same as in Fig. 1 but for Z=1, k=0.1, b=0.1, µ=1.

where Iν is the modified Bessel function [24]. Inserting (37) into the integral (34), we get

qsn(k,ξ)=

(−1)n

√2b

1∫

0

dxxit−1

(1− ikb −x)

(

1+ ikb −x

1− ikb −x

)n

exp

(

ikξx

1− ikb −x

)

×Ln

[

2kb kξx

(1− ikb −x)(1+ ik

b −x)

]

.

(38)

Expanding the exponential factor in (38) qsn can be expressed in the form

qsn(k,ξ)=

e−bξ

√2b

1+ω

2ω

∞

∑m=0

(bξ)m

m!

[

n

∑ℓ=0

(−1)ℓ(

n

ℓ

)

(2bξ)ℓ

ℓ!

n−ℓ

∑j=0

(−1)j

(

n−ℓ

j

)(

1+ω

ω

)j

×Γ(m+ j+1)Γ(it+ℓ)

Γ(it+ℓ+m+ j+1)2F1

(

it−ℓ,m+ j+1; it+ℓ+m+ j+1;1+ω

2ω

)

]

.

(39)Assuming that there exist a factor xǫ, ǫ=+0 in the integrand, we can perform the integral(38) by parts

qsn(k,ξ)=

1

2ktψn(ξ)−

(−1)n

it√

2b

1∫

0

dxxit

× d

dx

(

1+ ikb −x

)n

(

1− ikb −x

)n+1exp

(

ikξx

1− ikb −x

)

Ln

[

2kb kξx

(1− ikb −x)(1+ ik

b −x)

]

.

(40)


0 10 20 30 40 50

-0,5

0,0

0,5

Z = 1; k = 0.1; b = 0.1;

n = 0 n = 1 n = 5 n = 10 n = 15 n = 20

QnIm

Figure 4: The same as in Fig. 3 but for the imaginary parts.

We have found in our numerical calculations that it is preferable to use this represen-tation rather than the series (39).

Note that the Green’s function

g(+)(

k; ξ,ξ′)

=Γ(it)

21F1(it, 1,−ikξ<) [U(it, 1;−ikξ>)

− α1F1(it, 1,−ikξ>)] eikξ ′ ,

(41)

also meets the equation (32). In particular, with

α= e−πtΓ(1−it) (42)

the asymptotic behavior of the function

qn(k,ξ)≡∫

dξ′g(+)(k; ξ,ξ′)ψn(ξ′)=qs

n(k,ξ)−α1F1(it, 1,−ikξ)vn, (43)

in view of the asymptotic formulae

1F1(it, 1,−ikξ) ∼ξ→∞

eπt

Γ(1−it)(−ikξ)−it

(

1− t2

ikξ

)

+e−ikξ

Γ(it)(−ikξ)it−1, (44)

U(it, 1,−ikξ) ∼ξ→∞

(−ikξ)−it

(

1− t2

ikξ

)

, (45)

becomes

qn(k,ξ) ∼ξ→∞

vne−

πt2

ik

Γ(1−it)

Γ(it)

e−ikξ+itlnkξ

ξ. (46)


These (decaying) functions (43) serve as a basis for the quasi-Sturmian construction.We note only that the Green’s function G(+) which satisfies the equation

h(k)G(+)(k; ξ,ξ′)=δ(ξ−ξ′), (47)

h(k)=1

µξ

[

−2∂

∂ξξ

∂

∂ξ−2ikξ

∂

∂ξ+2kt

]

, (48)

is related to (41) by

G(+)(k; ξ,ξ′)=µg(+)(k; ξ,ξ′)ξ′. (49)

Thus, it follows from the recurrence relation

ξψn =1

2b[−(n+1)ψn+1+(2n+1)ψn−nψn−1] (50)

that the one-dimensional quasi-Sturmian of Eq. (31) are expressed in terms of qn (43)

Qn =µ

2b[−(n+1)qn+1+(2n+1)qn−nqn−1]. (51)

In Figs. 1-4 we present the plots of various QS functions for different parameters andenergies used in our calculations below.

3.2 Laguerre representation of the Quasi Sturmian functions

3.2.1 Expressions for the qn and Qn functions

The coefficients in the basis set (21) expansion of the Green’s function (33)

gs(+)(k; ξ,ξ′)=∞

∑m,n=0

ψm(ξ)gs(+)mn (k)ψn(ξ) (52)

are expressed in terms of well-known special functions

gs(+)mn (k)= i

2kω−1ωn+1 pν(t; ω)q

(+)µ (t; ω),

ν=min(n,m),µ=max(n,m),(53)

where

pn(t; ω)=(−1)n

n!

Γ(n+1−it)

Γ(1−it)2F1(−n, it; it−n; ω), (54)

q(+)n (t; ω)=(−1)n n!Γ(it)

Γ(n+1+it)2F1

(

it,n+1; n+1+it; ω−1)

. (55)

Then, it follows from (41), (36), (57), (56) and the expression

an =

√

2

b

(

1+ω

2

)it

(−1)n Γ(n+1−it)

n!Γ(1−it)2F1(−n, it; it−n; ω). (56)


0 10 20 30 40 50

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0p

1 = 1; p

3 = 0.1

N = 20 N = 25 N = 30 N = 35 N = 40 N = 45

scRe

Figure 5: Convergence of the equation (70) solution real part vs the number N of the basis quasi-Sturmiansused in the calculation (for each of the coordinates ξ1 and ξ3) on the diagonal ξ1= ξ3.

for the coefficients of the Kummer function expansion

1F1(it, 1,−ikξ)=∞

∑n=0

anψn(ξ), (57)

thatg(+)mn (k)= g

s(+)mn (k)−αam vn. (58)

Finally, from (31) it follows that

Qn(k,ξ)=∞

∑m=0

G(+)mn (k)ψm(ξ), (59)

whereG(+)mn =

µ

2b

[

−(n+1)g(+)mn+1+(2n+1)g

(+)mn −ng

(+)mn−1

]

. (60)

3.2.2 Expansion for the derivatives of Qn

To obtain the coefficients for the expansion

d

dξQn(k,ξ)= ∑

m=0

Cmn(k)ψm(ξ) (61)

let us calculate the integral

Cmn(k)=

∞∫

0

ψm(ξ)d

dξQn(k,ξ) (62)


by parts

∞∫

0

ψm(ξ)d

dξ Qn(k,ξ)=−ψm(0)Qn(k, 0)−∞∫

0

[

ddξ ψm(ξ)

]

Qn(k,ξ)

=−√

2bQn(k, 0)+b

[

∞∫

0

ψm(ξ)Qn(k,ξ)+2m−1

∑ℓ=0

∞∫

0

ψℓ(ξ)Qn(k,ξ)

]

.

(63)

From (59) it follows that

Cmn(k)=−√

2bQn(k, 0)+b

[

G(+)mn (k)+2

m−1

∑ℓ=0

G(+)ℓn (k)

]

. (64)

Similarly, we obtain that the coefficients dn of the derivative of (57) expansion

d

dξ1F1(it, 1,−ikξ)=

∞

∑n=0

dnψn(ξ), (65)

are

dn =−√

2b+b

[

an+2n−1

∑m=0

am

]

. (66)

4 A model problem

4.1 Statement of the problem

As mentioned above, the full three-body problem is very complicated. So we decided,first, to test the methodology by solving a simpler problem. We found one model prob-lem which contains many characteristics of the full one. This was introduced in 1997 inthe treatment of a double continuum wave function for two electrons interacting with aheavy ion [21]. This is actually an extension of the Φ2 model designed to deal with twoheave ions and one electron [25]. In Ref. [21] the authors introduced an approximation tothe full two-electron Schrodinger equation. This equation was obtained by demandingvery particular kinematical conditions on the system. The equation to be solved is

[

h1(k1)+h3(k3)−8k3

k1

∂2

∂ξ1∂ξ3

]

Ψ(ξ1,ξ3)=0. (67)

In this contribution we solve this equation using our QS functions.To start with the study, we divide the wave function Ψ into two parts

Ψ(ξ1,ξ3)=ΨC2(ξ1,ξ3)+Ψsc(ξ1,ξ3), (68)

where

ΨC2(ξ1,ξ3)= 1F1

(

iµ1Z1

k1, 1,−ik1ξ1

)

1F1

(

iµ3Z3

k3, 1,−ik3ξ3

)

. (69)


0 10 20 30 40 50

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

p1 = 1; p

3 = 0.1

N = 20 N = 25 N = 30 N = 35 N = 40 N = 45

scIm

Figure 6: The same as in Fig. 5 but for the solution imaginary part.

0 10 20 30 40 50

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

p1 = 1; p

3 = 0.1 N = 20

N = 25 N = 30 N = 35 N = 40 N = 45

scRe

Figure 7: The same as in Fig. 5 but on the axis ξ1.

This transforms Eq. (67) into the driven equation[

h1(k1)+h3(k3)−8k3

k1

∂2

∂ξ1∂ξ3

]

Ψsc(ξ1,ξ3)=8k3

k1

∂2

∂ξ1∂ξ3ΨC2(ξ1,ξ3). (70)

The scattering function Ψsc(ξ1,ξ3) is assumed to have purely outgoing behavior and canbe expressed as a finite series of the QS functions (31) products

Ψsc(ξ1,ξ3)=N−1

∑n1,n3=0

cn1 n3Qn1(p1,ξ1)Qn3(p3,ξ3). (71)


0 10 20 30 40 50

-0,2

0,0

0,2

0,4

0,6

p1 = 1; p

3 = 0.1 N = 20

N = 25 N = 30 N = 35 N = 40 N = 45

scIm


Note that in (71) it is not necessary that the parameters pj be equal to the momenta kj.Rather than use two-dimensional functions obtained from the action of the separablepart of Eq. (67), we propose here an expansion in products of one-dimensional functions.This is one of the aims of the paper, to see whether this expansion works and whether itis efficient.

Inserting the expansion (71) into (70) and projecting onto ψm1(ξ1)ψm3(ξ3) gives, in

0 10 20 30 40 50

-20

-16

-12

-8

-4

0

4p

1 = 1; p

3 = 0.1

N = 20 N = 25 N = 30 N = 35 N = 40 N = 45

scRe



0 10 20 30 40 50

-12

-8

-4

0

4

8

12

16p

1 = 1; p

3 = 0.1

N = 20 N = 25 N = 30 N = 35 N = 40 N = 45

scIm


view of the evident relation

h(k)= h(p)− 2i

µ(k−p)

∂

∂ξ, (72)

the following system of linear equations for the unknown coefficients cn1 n3

N−1

∑n1,n3=0

δm1 n1G(3)(+)m3n3

(p3)+G(1)(+)m1 n1

(p1)δm3 n3−[

2i

µ1(k1−p1)C

(1)m1n1

(p1)

×G(3)(+)m3 n3

(p3)+G(1)(+)m1n1

(p1)2i

µ3(k3−p3)C

(3)m3 n3

(p3)

+8k3

k1C(1)m1n1

(p1)C(3)m3n3

(p3)

]

cn1 n3

=8k3

k1d(1)m1

d(3)m3

, 0≤m1,m3≤N−1,

(73)

where C(j)m,n and d

(j)m are defined by (64) and (66), respectively.

4.2 Results

We follow Ref. [21] and put Z1=−2, µ1=1, k1 =1, and Z3=1, µ3=12 , k3 =0.4. Intuitively

one expects that convergence can be achieved provided that the sum of the first two(‘unperturbed’) terms in the figure brackets on the left-hand-side of Eq. (73) appears tobe much larger than the (‘perturbation’) term in the square brackets. We believe that itshould be possible to affect the ratio between these two parts of the matrix element byvarying the basis parameters pj.


Our calculations show that the convergence rate and numerical stability may be sig-nificantly improved by taking appropriate values of p1 and p2. The results obtained withparameters p1=1 and p3=0.1 (the Laguerre scale factors bj are equal to the pj) are shownin Figs. 5-10 where we plot the real and imaginary parts of the scattering wave functionΨsc on the diagonal ξ1 = ξ3 and the axes ξ1 and ξ3. We can see in all the figures that theconvergence is achieved showing that the method works. Results for N = 45 are shownin Figs. 11, 12.

0

10

20

30

40

50

-15

-10

-5

0

5

0

10

20

3040

50

scRe

Figure 11: The solution Ψsc to Eq. (70) (obtained with N=45) real part.

0

10

20

30

40

50

-15

-10

-5

0

5

10

15

20

0

10

20

3040

50

Im sc

Figure 12: The same as in Fig. 11 but for the imaginary part.


5 Conclusions

In this contribution we presented a study of the three-body scattering problem written inparabolic coordinates. It is well known that the C3 wave function [14] possesses the cor-rect asymptotic behavior in the Ω0 region where all the particles are far from each other.Thus, this is a good starting point to formulate a Lipmann-Schwinger type equation or adriven equation. This means, if we consider the C3 function as the asymptotic solution,the scattering part (the remaining part of the solution) will satisfy an equation having acompact kernel or a short range driven term. Due to the character of the perturbationcorresponding to the C3 function [26], the use of standard L2 basis is not appropriated.Instead, it turns to be necessary to use basis functions possessing the asymptotic behaviorcorresponding to the problem being considered. Generalized Sturmian [18] can be used.The numerical implementation of the calculation of this type of functions in paraboliccoordinates is part of our actual investigations. In this contribution we introduced a setof basis functions that we named quasi-Sturmian functions. They are defined differentlyfrom the generalized Sturmian as they are solutions of a driven differential equation. Weproposed here the use of QS functions which are the solutions of a driven equation whichincludes the separable part of the full three-body kinetic energy in the Klar’s paraboliccoordinates and also all the Coulomb interactions. In the right-hand-side of (30) any ba-sis set can be used. Depending on the type of the driven term appearing in the full threebody equation, the use of one or other of the bases can be more efficient. The basis onthe right-hand-side should be chosen to obtain the fastest convergence of the driven term.The QS functions form a complete basis set and this allows to expand the scattering wavefunction we are looking for. All the QS basis elements possess the correct asymptotic be-havior of the full three-body problem in the Ω0 region. This means that, in principle,only the inner region where the interaction between all the particles takes place must beexpanded.

In this contribution we tested the efficiency of the proposed method by applying it toa two-dimensional problem which possesses most of the properties of the full problem:the non-separability and the scattering type boundary conditions. Instead of using non-separable two-dimensional QS functions, products of one-dimensional ones in terms ofthe parabolic coordinates appearing in the model problem were utilized. We showedthat convergence is reached for the scattering wave function with a reasonable numberof basis functions. A more extensive study of the properties of the QS functions for basesof different types in the driven terms will be presented soon elsewhere.

Acknowledgments. One of the authors (GG) thanks the support by PGI (24/F049)of the Universidad Nacional del Sur. GG acknowledge also the support of ANPCyT(PICT08/0934) (Argentina) and PIP 200901/552 CONICET (Argentina).


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