Define a fundamental solution of a linear partial differential operator $P$. Prove that the function

$$G(x)=\frac{1}{2} e^{-|x|}$$

defines a distribution which is a fundamental solution of the operator $P$ given by

$$P u=-\frac{d^{2} u}{d x^{2}}+u .$$

Hence find a solution $u_{0}$ to the equation

$$-\frac{d^{2} u_{0}}{d x^{2}}+u_{0}=V(x),$$

where $V(x)=0$ for $|x|>1$ and $V(x)=1$ for $|x| \leqslant 1$.

Consider the functional

$$I[u]=\int_{\mathbb{R}}\left\{\frac{1}{2}\left[\left(\frac{d u}{d x}\right)^{2}+u^{2}\right]-V u\right\} d x .$$

Show that $I\left[u_{0}+\phi\right]>I\left[u_{0}\right]$ for all Schwartz functions $\phi$ that are not identically zero.