Delta function with different variables

Question

How can I deal with something like that:

$\int\int\int dx dydz \delta(E-E_0+x^2+y^2+z^2)$

I could substitue $x^2\rightarrow a$ and do the first integral, but the the delta function vanishes?

Best

Answer 1

With the exception of $E=E_0$, this just means you integrate over the subspace, where $E-E_0+x^2+y^2+z^2=0$. This can be done with various methods: you can find coordinates where this is trivially satisfied (spherical coordinates, for instance), or you could just express one of the variables with the rest (accounting for multiple branches) and integrate that way.

Answer 2

First, we note that if $E>E_0$, then $E-E_0+x^2+y^2+z^2>0$ and thus the integral is zero.

Now, assume that $E<E_0$. Use of spherical coordinates will facilitate analysis. So, let $r^2=x^2+y^2+z^2$. Then, we have

$$\begin{align} \iiint\delta (E-E_0+x^2+y^2+z^2)dx\,dy\,dz&=4\pi\int_0^{\infty}r^2\delta \left(r^2-(E_0-E)\right)dr\\\\ &=4\pi\int_0^{\infty}\,\left(u+(E_0-E)\right)\frac{\delta \left(u\right)}{2\sqrt{u+(E_0-E)}}du\\\\ &=2\pi\sqrt{E_0-E} \end{align}$$