Method of Hyperspherical

Functions

Roman.Ya.Kezerashvili

New York City Technical CollegeThe City University of New York

Objectives •Differential Equations in 3- 6- and 9- dimensional Spaces.

•Hyperspherical Functions•Asymptotic Behavior of the Solutions of These Equations

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

3-D Universe ?!

r

z

yx

)()()(1

)(1

22

2rErrV

rrr

rr

The second order linear differential equation for eigenvalues and eigenfunction

)()()(2 rErrV

For Euclidean 3-D space and a rectangular coordinate system

cos

sinsin

cossin

z

ry

rxSpherical coordinate

kz

jy

ix

Gradient

Separation of Variables

Assume a solution in the form

),()(

)(0

lml

lm

lm

l Yr

rur

),()1(),(

operator Laplace theofpart angular theof

ion eigenfunct theis harmonic) (spherical ),(

lmlm

lm

YllY

Y

0)()()1(

)(22

2

rurV

r

llEru

dr

dll

The second order linear differential equation for eigenvalues and eigenfunction

Differential Equation in 6-D Differential Equation in 6-D SpaceSpace

We introduce the Jacobi coordinates, defined by

x2

x1

1

23

,232

2/1

32

321 rr

mm

mmx

,

212

32

3322

2/1

321

3212

rrmm

mrrm

mmm

mmmx

),(),(),( 21212122

21xxExxxxVxx

Equation for three body in Euclidean 3-D space and a rectangular coordinate system

21

21

222

2

22

222

sin

1

cos

12cot4)(

where)(15

xx

xx

K

Kd

d

d

d

Let us introduce hyperspherical functions K as eigenfunctions of the angular part of the six dimensional Laplace operator

numberinteger positive a is ),()4()()( KKKKKK

),,( .sin ;cos ; 211122

21

2 xxxxxx

Let us introduce hyperspherical coordinate in Euclidian Six dimensional space as

Let expand the function by a complete set of hyperspherical functions ),(21

xx

)()(

),(221

KK

Ku

xx

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

)()()()2()(1)(

2

2

2

2

K

KKKK

KK uWuK

Ed

du

d

ud

We shell seek the solution of this system of differential equations in the form

functionNeumann and Bessel from dconstructe

matrices diagonal are )( and )( where

)()()()()()( 11

NJ

ATVNVJu

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

0

11 )()()()()()()()(2

exp)()( UTUNUJWNdUA

Amplitude function

)()()()()(

)()()()()()(2

)()(

)()(

11

1

1

TUNUJ

WNUTJUUd

dUT

d

dT

Nonlinear system of differential equations for phase functions

The Asymptotic Behavior of Elastic 2->2 Scattering Wave Function

The process 2->2The process 2->2

Plane wave in 3-D configuration space

Spherical wave in

3-D configuration

space

The asymptotic wave function

)()()( 220 krUkrkr

eation spacD configurin

plane waveekr rki

3

)(0

eation spacD configurin

l wave sphericar

e)kr(U

irk

3

22

.)]()][(1[),(

)(

);,(1)()(

)2(),,(

*22'2

'

2'22'

22''

222

3

2121

2121

2121

ommll

K

Kmmll

KmmlKl

Kd

qiopk

pkt

ioqp

ioqkptJqdpdiqpyxU

The wave function describing the 3->3 process asymptotically behaves as

),,(),,(),,(),,(),,( 33

3

10 qpyxUqpyxUqpyxUqpyxqpyx ds

*0

2211

2121

221122

)( )]([)()()(

2

mlmlK

mmlKl

mlmlKK

Kqypxi Jie Plane wave in 6-D

configuration space

,)]([),(1

)()()2(

),,( *22'

'

222

3 21212121

2121

ommll

KKmmll

KmmlKl

Ks

iopp

qptJpdiqpyxU

Single scattering

1

23

Double scattering

1

2

3

2

1

3

Asymptotic Behavior

iks eU

2/5

1

ikD eU

2/5

1

Single scattering

Double scattering

Optical TheoremThe Optical Theorem gives the relationship between a total cross section and imaginary part of a forward scattering amplitude

3-D Space

6-D Space

)0(Im4

2F

k

Lll

dFk

21

2/

0

222

2

cossin)0(Im)4(