= Solution
A <fundamental solution> of a linear <differential operator> $P$ is a <distribution> $G$ satisfying $PG=\delta_0$. The locally integrable function $G=e^{-|x|}/2$ defines a <distribution> by integration against compactly supported smooth test <functions>. Away from zero it satisfies $-G''+G=0$. It is continuous at zero, while $G'(0+)=-1/2$ and $G'(0-)=1/2$. Consequently its distributional second derivative is its ordinary derivative on the two sides plus the derivative jump times $\delta_0$:
$$
G''=G-\delta_0,\qquad\boxed{(-d^2/dx^2+1)G=\delta_0.}
$$
This derivative-jump formula follows by <integration by parts> on each half-line; the boundary terms equal $-\phi(0)$ in $G''$.
<Convolution> with the indicator forcing gives
$$
u_0(x)=\frac12\int_{-1}^1e^{-|x-y|}\,dy
=\boxed{\begin{cases}1-e^{-1}\cosh x,&|x|\leq1,\\
\sinh(1)e^{-|x|},&|x|\geq1.\end{cases}}
$$
The two values and their first derivatives agree at $\pm1$, so no delta terms arise there. This is a decaying <weak solution>, with the equation holding classically away from the forcing jumps.
For real Schwartz $\phi$, expand the <energy>:
$$
I[u_0+\phi]-I[u_0]
=\int(u_0'\phi'+u_0\phi-V\phi)\,dx
+\frac12\int[(\phi')^2+\phi^2]\,dx.
$$
The mixed term is zero by the weak equation $-u_0''+u_0=V$, or <integration by parts> using the continuous first derivative and decay. The remaining integral is strictly positive whenever $\phi$ is not identically zero. Hence
$$
\boxed{I[u_0+\phi]>I[u_0]\quad(0\ne\phi\text{ a real Schwartz function}).}
$$
The real-valued function space is implicit in the stated ordered quadratic functional; for complex <functions> one instead uses absolute squares and the corresponding real part of the forcing term.
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