Method of Hyperspherical Functions

1. Method of Hyperspherical Functions Roman.Ya.KezerashviliNew York City Technical CollegeThe 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 ?! Time Space Matter Symmetry

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

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

7. 1 x1 x2 3 2 Differential Equation in 6-D Space We introduce the Jacobi coordinates, defined by

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

9. 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 Let expand the function by a complete set of hyperspherical functions We shell seek the solution of this system of differential equations in the form

10. Nonlinear system of differential equations for phase functions Amplitude function 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

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

12. The asymptotic wave function

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

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