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