Equation (4.24) represents a Fredholm integral equation of second kind with a free term determined by function
. This equation can
be rewritten4.4:
![$\displaystyle f(x)=\int f(x^{'})K(x^{'},x)\,dx^{'}+g(x),$](img888.png) |
(4.25) |
where the
and the free term are given functions. The multidimensional variable
stands for
4.5. The resolvent series4.6 of a Fredholm integral equation is obtained by replacement of its right hand side into itself:
![$\displaystyle f(x)=g(x)+\int g(x_{1})K(x_{1},x)\,dx_{1}+\int dx_{1}\int dx_{2}\,g(x_{1})K(x_{1},x_{2})K(x_{2},x)+\cdots$](img892.png) |
(4.26) |
This means that the solution of (4.24) can be written as consecutive iterations of the free term:
![$\displaystyle f(x)=f^{0}(x)+f^{1}(x)+f^{2}(x)+\cdots.$](img893.png) |
(4.27) |
To find the iteration terms explicitly (4.24) is rewritten as:
where
has been replaced by
and the respective quasi-momentum space trajectory is:
Introducing a quasi-momentum space trajectory for the time interval
:
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![$\displaystyle \vec{K}_{1}(t_{2})=\vec{k}_{1}-\frac{q}{\hbar}\vec{E}_{s}(t_{1}-t_{2}),$](img902.png) |
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![$\displaystyle \vec{K}_{1}(t_{1})=\vec{k}_{1},$](img903.png) |
(4.30) |
one obtains for
:
Substituting (4.32) into (4.29) gives:
From (4.34) the first iteration term is obtained:
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![$\displaystyle f_{1}^{(1)}(\vec{k},t)=$](img912.png) |
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![$\displaystyle \int_{0}^{t}\,dt_{1}\int\,d\vec{k}_{1}G(\vec{K}_{1}(0))\exp\biggl...
...(t_{1}))\exp\biggl(-\int_{t_{1}}^{t}\widetilde{\lambda}[\vec{K}(y)]\,dy\biggr).$](img913.png) |
(4.33) |
which is also schematically shown in Fig. 4.1.
Figure 4.1:
Graphical representation of the first iteration term.
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In order to obtain the second iteration term the third quasi-momentum space trajectory is introduced for
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![$\displaystyle \vec{K}_{2}(t_{3})=\vec{k}_{2}-\frac{q}{\hbar}\vec{E}_{s}(t_{2}-t_{3}),$](img916.png) |
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![$\displaystyle \vec{K}_{2}(t_{2})=\vec{k}_{2}.$](img917.png) |
(4.34) |
Then for
in (4.32) one obtained:
The second iteration term is obtained from (4.36) by replacing
with the free term of (4.29):
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![$\displaystyle f_{1}^{(2)}(\vec{k},t)=\int_{0}^{t}\,dt_{1}\int\,d\vec{k}_{1}\int...
...exp\biggl(-\int_{0}^{t_{2}}\widetilde{\lambda}[\vec{K}_{2}(y)]\,dy\biggr)\times$](img923.png) |
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![$\displaystyle \times\widetilde{S}(\vec{k}_{2},\vec{K}_{1}(t_{2}))\exp\biggl(-\i...
...(t_{1}))\exp\biggl(-\int_{t_{1}}^{t}\widetilde{\lambda}[\vec{K}(y)]\,dy\biggr).$](img924.png) |
(4.36) |
This term is displayed graphically in Fig. 4.2.
Figure 4.2:
Graphical representation of the second iteration term.
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S. Smirnov: