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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;
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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
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12
3
x1y1
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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
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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
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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.
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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
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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
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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
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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)
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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
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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
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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 |
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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
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He-He bound state
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Chebyshev expansionof v * Psi for He-He bound state
3.5 < r < 40
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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
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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
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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
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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.