Solution of Faddeev Integral Equations in Configuration Space Walter Gloeckle, Ruhr Universitat...
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Transcript of Solution of Faddeev Integral Equations in Configuration Space Walter Gloeckle, Ruhr Universitat...
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.