Solution of Faddeev Integral Equationsin

Configuration Space

Walter Gloeckle, Ruhr Universitat BochumGeorge Rawitscher, University of Connecticut

Fb-18, Santos, Brazil, August 24, 2006

Work in Progress

physics/0512010;

AIM: Solve three-body problems for Atomic Physics

Method:

1.Use Faddeev Equations in Configuration space

2.Use only integral equations for the productpotential x Wave function, called T

3.Numerical discretization via theSpectral expansion in terms of Chebyshev Polynomials

12

3

x1y1

Two-BodyThree-Body

Tr Vr r

0

E; r, r r dr

T1 t1 2 3

t i V i V iG0t i, i 1,2,3

1 1 G0 T1 2 G0 T2 3 G0 T3

V V g0

T = Product of wave function times potentialt or t - matrix

x y| tE |xy 12 3

dq e iqy yx|Eq|x

Eq E q2

2Mi

Two-b T-matrix imbedded in three-b space Two-body

G0E 1E iH0

x y|G0|x y 12 3

dq e iqy yg0x,x ;Eq

Three-body free Green’s function

Two-body free Green’s function

1x1,y1 2x2,y2 3x3,y3

x y V ix i E ixi,yi

V ixi jx j ,yj kxk ,yk

1 1 G0 t1 2 3 2 G0 t2 3 1 3 G0 t3 1 2

T1 t1G0T2 T3

T2 t2 1 t2G0T3 T1

T3 t3 1 t3G0T1 T2

Differential Fad’v Eq.for the wave fctn.

Integral Fad’v Eqfor the wave fctn.

Integral Fad’v Eqfor the T - fctn.

Coupled Faddeev Eqs.

With 3b-Pot’l

T1 t1G0T2 T3 1 t1G0 V41 1 G0 T1 T2 T3

T2 t2 1 t2G0T3 T1 1 t2G0 V42 1 t2G0T3 T1

1 t2G0V42G0T1 T2 T3

T3 t3 1 t3G0T1 T2 1 t3G0 V43 1 t3G0T1 T2

1 t3G0V43G0T1 T2 T3

A big mess, that requires the two-body t-matrices ti

I = 1, 2, 3

FE; r sinkr;GE; r coskr or expikr

Two-b tau-matrix, one dimensionTwo variables

E; r, r Vr r r RE; r, r

RE; r, r Vr g0E; r, r Vr

Vr 0

g0E; r, r RE; r, r dr

g0r, r 1kFr Gr

R r i, r j A ir j Yir i B ir j Zir i, i j

R r i, r j Air j Yir i B ir j Zir i, i j

Spectral Integral Equation Method

1 2 i j

Partitions

Result: Obtain a Rank 2 separable expression

0 < r < 3000 a.u.

He-He binding energy via the S-IEM

Tol 103 a0 1 No.of Partitions No. of Meshpts

10 12 5.0817542 47 799

10 6 5.0817461 19 323

10 3 5.0776 13 221

Rawitscher and Koltracht, Eur. J. Phys. 27, 1179 (2006)

Computing time for MATLAB (sec) with S-IEM

2.8 GHz Intel computer,200 Partitions, 17 points per partitionS-IEM

sec

r 1/10

r, r 2

Next Steps: toy model

2. Ignore the three-body interaction, and solve for identical particles

x y| tE |xy 12 3

dq e iqy yx|Eq|x

Eq E q2

2Mi

1. Go to the configuration representation

1 P 1 1 1 G0t1P 1 1 1 G0T

T t1P 1 t1PG0T

3. Make a partial wave exp.; set all L= 0

Kx,y; x,y 2

0

dq q2j0qyj0qy0x,x; q

0x,x; q 0

dxx2 0x,x; qg0x,x; q

dx,y 2

2 433

0

dq q2j0qy

0

dyy2j0qy

0

dy y 2j0q0y

1

1dt r0x, | 43 y

23y |; qun0|

23y 4

3y |

1

2 Vx

1

1dt un0|

12x y| j0q0| 34 x 1

2y|

Tx,y dx,y

0

0

dxdyxy2 Kx,y ; x,y Tx,y

Tx,y 1

1dt T| 1

2x y |, | 3

4x 1

2y |

Ansatz:

Tx,y s,r 1nL asr Lsx Lry

Tx,y s,r 1nL asr Fsrx,y

Fsrx,y 1

1Ls| 12 x

y | Lr| 34 x 1

2y | dt

Basis Functions

He-He bound state

Chebyshev expansionof v * Psi for He-He bound state

3.5 < r < 40

s,r 1nL ast Lsx Lry dx,y s,t 1

nL asrK srx,y

K srx,y Kx,y ; x,y Fsrx,y dx dy.

Equations for the expansion coefficients

I Ma d

Msr;sr Lsx Lry K srx,y dx dy

Final Matrix eqs.

dsr Lsx Lry dx,y dx dy

Complexity Estimates

nx ny 200

nL 100n t 100;nP nq 50.

nI 10

# of coordinate points

# of basis functions

# of angles

# of partitions and q values

Additional computational factor

F srx 0

ymaxj0qyFsrx,ydy

nF ny nx nL2 nq nI 2 1011 FLOPs

Msr,sr

nM nL2 nL

2 nq nP 2.5 1011 FLOPs

#

#

Ingredients for the Toy Model matrix eq.

Solution of the matrix eq.

nL23 1012 FLOPs

1-2 Hours Fortran

on a 2 GHz PC

Summary and Conclusions

• Integral Faddeev Eqs. in Config. Space for T(x,y) = V x Psi, combined with the spectral method for solving integral equations;

• Greens function incorporate asymptotic boundary conditions;

• Toy model should take about one hour• The expected accuracy is more than

6 sign. figs.