Download Method of Hyperspherical Functions

Method of
Hyperspherical
Functions
Roman.Ya.Kezerashvili
New York City Technical College
The City University of New York
Objectives
•Differential Equations in 3- 6- and 9dimensional 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 ?!
For Euclidean 3-D space and a rectangular coordinate system





   V (r ) (r )  E(r )
2
     
 i 
j k
x
y
z
r
Gradient
z
Spherical coordinate
x  r sin q cos f
y  r sin q sin f
z  cos f
x
q
f
The second order linear differential equation for eigenvalues and
eigenfunction
  

1
1  2 
 r 2 r (r r )  r 2 q  V (r )  (r )  E (r )
y
Separation of Variables
Assume a solution in the form
m l

ul ( r )
 (r )   
Ylm (q , f )
r
l 0 m   l
Y (q ,f ) (spherical harmonic) is the eigenfunction
lm
of the angular part of the Laplace operator 
qf
 Y (q ,f )  l (l  1)Y (q ,f )
qf
lm
lm
The second order linear differential equation for eigenvalues and
eigenfunction
d
l (l  1)


u (r )   E 
 V (r )u (r )  0
dr
r


2
2
l
2
l
Differential Equation in 6-D Space
1
We introduce the Jacobi coordinates, defined by
x2
1/ 2
  2m2 m3 

x1  
 m2  m3 
3
 
r2  r3 ,
  2m1 m2  m3  

x2  
 m1  m2  m3 
1/ 2
 
 m2 r2  r3 m3   

r2  r1 ,
 m2  m3

x1
2

Equation for three body in Euclidean 3-D space and a rectangular
coordinate system

 
 
 
     V ( x1 , x2 ) ( x1 , x2 )  E( x1 , x2 )
2
x1
2
x2
Let us introduce hyperspherical coordinate in Euclidian Six
dimensional space as
 
 2  x12  x22 ; x1   cos ; x1   sin  .   ( , x1 , x2 )
 d2

5 d
1 
 
 


K
(

)
 where
2
2
 d

 d


2

1
1
K ( )  

4
cot
2




 x2
x1
2
2
2


cos 
sin 
2
x1
2
x2
Let us introduce hyperspherical functions FK as eigenfunctions of
the angular part of the six dimensional Laplace operator

K ()F ()  K ( K  4)F (), K is a positive integer number
K
K
 
Let expand the function  ( x , x ) by a complete set of hyperspherical functions
1
2
u ()
 
( x , x )  
F ()

K
1
2
K

K
2
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
d u (  ) 1 du (  ) 
( K  2)

 E 
d
 d


2
K
K
2
2
2

u (  )  W (  )u (  )

K
K
KK 
K
We shell seek the solution of this system of differential equations in the form


u (  )  J (  )V 1 (  )  N (  )V 1T (  ) A(  )
where J (  ) and N (  ) are diagonal matrices
constructe d from Bessel and Neumann 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
Nonlinear system of differential equations for
phase functions
dT (  ) 
dU (  )


 T (  )
U (  )    U (  ) J (  )  T (  )U (  ) N (  )W (  ) 
d
d
2


1
1
  J (  )U (  )  N (  )U (  )T (  ) 
1
1
Amplitude function
 

A(  )  U (  ) exp  dN (  )W (  )  J (  )U (  )  N (  )U (  )T (  )U (  )
 2


0
1
1
The Asymptotic Behavior of Elastic 2->2 Scattering
Wave Function
The process 2->2
Spherical wave in
3-D configuration
space
Plane wave in 3-D
configuration space
The asymptotic wave
function






(r k )  F0 (r k )  U 22 (r k )


ik r
F 0 (r k )  e  plane wave
in 3  D configuration space

eirk
U 22(r k ) 
spherica l wave
r
in 3  D configuration space
The wave function describing the 3->3 process
asymptotically behaves as
3
  
  
   

  
d   
( x, y p, q )  F 0 ( x, y p, q )  Us ( x , y p , q )   U
( x , y p , q )  U 33 ( x, y p, q )
 1
 
2
l1m1l2 m2
l1m1l2 m2
K
*
i
J
(

)
F
(

)
[
F
(

)]

K

2
0
K
K
( ) 2 Kl1l2m1m2
  
e i ( x  p  y q ) 
   
(2 )3
U ( x , y p , q )  
2
s

i
K
Kl1l2 m1m2
1
F
l1l2 m1m2
K
Plane wave in 6-D
configuration space
 

1 t ( p' , q )
l1l2 m1m2
*
()  dp J K  2 ( ) 2 '2
[
F
(

)]
,
K
o
2
 p  p  io
Single
scattering
2
3
 '
'
2
2
t
  
 
(2 )
1  ( p , k ;   q  io )
l1l2 m1m2
d 
K
U ( x , y p , q ) 
 i F K () dp dq J K 2 ( )  2 p '2  ( 2  q '2 )  io
 2 Kl1l2 m1m2


3
1


'
t (k , p )
2
k
 p'2  io
[1   (q2 )][ F lK1l2 m1m2 ( o )]* .
Double 2
scattering 3
1
2
3
Asymptotic Behavior
U 
1
s
 
U
D

 

5/ 2
1

5/ 2
e
e
ik
Single
scattering
ik
Double
scattering
Optical Theorem
The Optical Theorem gives the relationship
between a total cross section and imaginary part of
a forward scattering amplitude
4
  2 Im F (0)
k
(4 )
 2
k
2
 Im
l1l2 L
3-D Space
 /2
2
2
F
(
0
)
sin

cos
 d

0
6-D Space