How Does the Delta Function Simplify Integral Equations in Arken's Text?

[Hariraumurthy]

Homework Statement

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)

Homework Equations

I am stuck on equation 16.9. That is I am not sure how for the special case of v\left( {\vec r,\vec r'} \right) = v\left( {\vec r} \right)\delta \left( {\vec r - \vec r'} \right), that
\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) reduces to \left( {{\nabla ^2} + {a^2}} \right)\psi \left( {\vec r} \right) = v\left( {\vec r} \right)\psi \left( {\vec r} \right)
when

The Attempt at a Solution

If {\vec r} is in the region of integration \Omega (case 1), then using integration by parts, the reduced RHS is given by \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) since in general the integral of \delta \left( {\vec x} \right) over any region containing \vec x = 0 is 1. The second integral \int_\Omega ^{} d \left( {v\left( {\vec r'} \right)\psi \left( {\vec r'} \right)} \right) is just {\left[ {v\left( {r'} \right)v(r')} \right]_\Omega }. Therefore the RHS is 0 which is not the LHS.

Case 2: \vec r \notin \Omega. Doing the same integration by parts, the reduced RHS is
{\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.

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

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

Thanks in advance for replying.

 

[Ilmrak]

Hello,

You simply have to consider that

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

no integration by parts is needed :)

Ilm