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Integral equations problem.

  1. May 12, 2013 #1
    1. The problem statement, all variables and given/known data
    I am trying to read arken's section on integral equations because I need it for a problem I am trying to attack. I am stuck on a part of a page. I have attached the relevant excerpt from the book.(Not the whole book because it is copyrighted)


    2. Relevant equations

    I am stuck on equation 16.9. That is I am not sure how for the special case of [tex]v\left( {\vec r,\vec r'} \right) = v\left( {\vec r} \right)\delta \left( {\vec r - \vec r'} \right)[/tex], that
    [tex]\left( {{\nabla ^2} + {a^2}} \right)\psi \left( {\vec r} \right) = \int {v\left( {\vec r,\vec r'} \right)} \psi \left( {\vec r} \right){d^3}\left( {r'} \right)[/tex] reduces to [tex]\left( {{\nabla ^2} + {a^2}} \right)\psi \left( {\vec r} \right) = v\left( {\vec r} \right)\psi \left( {\vec r} \right)[/tex]
    when
    3. The attempt at a solution
    If [tex]{\vec r}[/tex] is in the region of integration [tex]\Omega [/tex] (case 1), then using integration by parts, the reduced RHS is given by [tex]\int\limits_\Omega ^{} {v\left( {\vec r'} \right)\psi \left( {\vec r'} \right)} \delta \left( {\vec r - \vec r'} \right){d^3}\left( {r'} \right) = {\left[ {v\left( {r'} \right)v(r')} \right]_\Omega } - \int_\Omega ^{} d \left( {v\left( {\vec r'} \right)\psi \left( {\vec r'} \right)} \right)[/tex] since in general the integral of [tex]\delta \left( {\vec x} \right)[/tex] over any region containing [tex]\vec x = 0[/tex] is 1. The second integral [tex]\int_\Omega ^{} d \left( {v\left( {\vec r'} \right)\psi \left( {\vec r'} \right)} \right)[/tex] is just [tex]{\left[ {v\left( {r'} \right)v(r')} \right]_\Omega }[/tex]. Therefore the RHS is 0 which is not the LHS.

    Case 2: [tex]\vec r \notin \Omega [/tex]. Doing the same integration by parts, the reduced RHS is
    [tex]{\left[ {v\left( {r'} \right)v(r')} \right]_\Omega }\int_\Omega ^{} {\delta \left( {\vec r - \vec r'} \right)} {d^3}\left( {r'} \right) - \int_\Omega ^{} {\left( {\left( {\int_\Omega ^{} {\delta \left( {\vec r - \vec r'} \right){d^3}\left( {r'} \right)} } \right)d\left( {v\left( {\vec r'} \right)\psi \left( {\vec r'} \right)} \right)} \right)} = 0 - 0 \ne RHS[/tex].

    In summary I am having trouble verifying that for the special case of 16.9, 16.8 reduces to 16.6.

    Also is [tex]\partial \Omega [/tex] fixed or not?(my guess is that the boundary is fixed because Arken transforms this in to a fredholm equation of the second kind later on in the page(included in the excerpt.)

    Thanks in advance for replying.
     

    Attached Files:

  2. jcsd
  3. May 12, 2013 #2
    Hello,

    You simply have to consider that

    [itex]\int_{\Omega}\mathrm{d}x f(x) \delta(x) = f(0) \; \mathrm{if} \; 0\in \Omega, \mathrm{or} =0 \; \mathrm{if} \; 0\notin\Omega[/itex];

    no integration by parts is needed :)

    Ilm
     
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