Let $\chi$ be a smooth cut-off function, defined on an interval $[0,\infty)$ and it is equal 1 on $[0,X-1)$ and goes to 0 on $[X-1,X)$ then equal 0 on $[X,\infty)$.. Let $v=exp(-ax)$ on $[0,\infty)$..
I need to find the $L^2$-norm of $\chi' v$ and here is my attempt:
$\| \chi' v \|^2=\int_{0}^{\infty} \chi'^2 v^2dx= \int_{X-1}^{X} \chi'^2 exp(-2ax) dx$ ..
Now, how can I finish the calculation because I don't know what is exactly $\chi'$ on $[X-1,X]$ ?
I appreciate any help ..