Solution of Faddeev Integral Equations in Configuration Space Walter Gloeckle, Ruhr Universitat Bochum George Rawitscher, University of Connecticut Fb-18, Santos, Brazil, August 24, 2006 Work in Progress physics/ ;

AIM: Solve three-body problems for Atomic Physics Method: 1.Use Faddeev Equations in Configuration space 2.Use only integral equations for the product potential x Wave function, called T 3.Numerical discretization via the Spectral expansion in terms of Chebyshev Polynomials

12 3 x1x1 y1y1

Two-Body Three-Body T = Product of wave function times potential t or  t - matrix

Two-b T-matrix imbedded in three-b space Two-body Three-body free Green’s function Two-body free Green’s function

Differential Fad’v Eq. for the wave fctn. Integral Fad’v Eq for the wave fctn. Integral Fad’v Eq for the T - fctn.

Coupled Faddeev Eqs. With 3b-Pot’l A big mess, that requires the two-body t-matrices t i I = 1, 2, 3

Two-b tau-matrix, one dimension Two variables

Spectral Integral Equation Method 12 i j Partitions Result: Obtain a Rank 2 separable expression

0 < r < 3000 a.u. He-He binding energy via the S-IEM 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 partition S-IEM

Next Steps: toy model 2. Ignore the three-body interaction, and solve for identical particles 1. Go to the configuration representation 3. Make a partial wave exp.; set all L= 0

Ansatz: Basis Functions

He-He bound state

Chebyshev expansion of v * Psi for He-He bound state 3.5 < r < 40

Equations for the expansion coefficients Final Matrix eqs.

Complexity Estimates # of coordinate points # of basis functions # of angles # of partitions and q values Additional computational factor

Ingredients for the Toy Model matrix eq. Solution of the matrix eq. 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.