6.4 Linear Systems 183
wheremis the reduced mass of the interacting particles. Furthemore, the interaction between
the particles,V, carries
In order to solve the Lippman-Schwinger equation in momentum space, we need first to
write a function which sets up the integration points. We need to do that since we are going
to approximate the integral through
∫b
a
f(x)dx≈
N
∑
i= 1
wif(xi),
where we have fixedNintegration points through the corresponding weightswiand points
xi. These points can for example be determined using Gaussian quadrature.
The principal value in Eq. (6.23) is rather tricky to evaluate numerically, mainly since com-
puters have limited precision. We will here use a subtraction trick often used when dealing
with singular integrals in numerical calculations. We use the calculus relation from the pre-
vious section ∫∞
−∞
dk
k−k 0
= 0 ,
or ∫∞
0
dk
k^2 −k^20
= 0.
We can use this to express a principal values integral as
P
∫∞
0
f(k)dk
k^2 −k 02
=
∫∞
0
(f(k)−f(k 0 ))dk
k^2 −k^20
, (6.26)
where the right-hand side is no longer singular atk=k 0 , it is proportional to the derivative
d f/dk, and can be evaluated numerically as any other integral.
We can then use the trick in Eq. (6.26) to rewrite Eq. (6.23) as
R(k,k′) =V(k,k′)+
2
π
∫∞
0
dq
q^2 V(k,q)R(q,k′)−k^20 V(k,k 0 )R(k 0 ,k′)
(k^20 −q^2 )/m
. (6.27)
We are interested in obtainingR(k 0 ,k 0 ), since this is the quantity we want to relate to experi-
mental data like the phase shifts.
How do we proceed in order to solve Eq. (6.27)?
- Using the mesh pointskjand the weightsωj, we can rewrite Eq. (6.27) as
R(k,k′) =V(k,k′)+
2
π
N
∑
j= 1
ωjk^2 jV(k,kj)R(kj,k′)
(k^20 −k^2 j)/m
−
2
π
k^20 V(k,k 0 )R(k 0 ,k′)
N
∑
n= 1
ωn
(k^20 −k^2 n)/m
. (6.28)
This equation contains now the unknownsR(ki,kj)(with dimensionN×N) andR(k 0 ,k 0 ).
- We can turn Eq. (6.28) into an equation with dimension(N+ 1 )×(N+ 1 )with an integration
domain which contains the original mesh pointskjforj= 1 ,Nand the point which cor-
responds to the energyk 0. Consider the latter as the ’observable’ point. The mesh points
become thenkjforj= 1 ,nandkN+ 1 =k 0.
- With these new mesh points we define the matrix
Ai,j=δi,j−V(ki,kj)uj, (6.29)
whereδis the Kroneckerδand
uj=
2
π
ωjk^2 j
(k^20 −k^2 j)/m
j= 1 ,N (6.30)
