1. Method of Hyperspherical Functions Roman.Ya.Kezerashvili New York City Technical College The City University of New York

2. Objectives Differential Equations in 3- 6- and 9- dimensional Spaces. Hyperspherical Functions Asymptotic Behavior of the Solutions of These Equations

3. The results are published in Journal of Mathematical Physics, 1983 Nuclear Physics 1984 Particles and Nuclei, 1986 Physics Letters 1993, 1994 Advances in Quantum Theory, 2001

4. 3-D Universe ?!

5. r z y x   The second order linear differential equation for eigenvalues and eigenfunction For Euclidean 3-D space and a rectangular coordinate system Spherical coordinate Gradient

6. Separation of Variables Assume a solution in the form The second order linear differential equation for eigenvalues and eigenfunction

7. Differential Equation in 6-D Space We introduce the Jacobi coordinates, defined by x2x2 x1x1 1 2 3

8. Equation for three body in Euclidean 3-D space and a rectangular coordinate system Let us introduce hyperspherical functions  K  as eigenfunctions of the angular part of the six dimensional Laplace operator Let us introduce hyperspherical coordinate in Euclidian Six dimensional space as

9. Let expand the function by a complete set of hyperspherical functions This expansion is substituted into previous equation and differential equation is separated into the system of differential equations for hyperspherical function and the system of second order differential equations for hyperradial functions We shell seek the solution of this system of differential equations in the form

10. Substituting this expression into the system of differential equations we obtain the nonlinear first order matrix differential equations for the phase functions and amplitude function Amplitude function Nonlinear system of differential equations for phase functions

11. The Asymptotic Behavior of Elastic 2->2 Scattering Wave Function The process 2->2 Plane wave in 3-D configuration space Spherical wave in 3-D configuration space

12. The asymptotic wave function

13. The wave function describing the 3->3 process asymptotically behaves as Plane wave in 6-D configuration space Single scattering 1 2 3 Double scattering 1 2 3 2 1 3

14. Asymptotic Behavior Single scattering Double scattering

15. Optical Theorem The Optical Theorem gives the relationship between a total cross section and imaginary part of a forward scattering amplitude 3-D Space 6-D Space