 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 $-G^{\prime \prime }+G=0$. It is continuous at zero, while $G^{\prime }(0+)=-1/2$ and $G^{\prime }(0-)=1/2$. Consequently its distributional second derivative is its ordinary derivative on the two sides plus the derivative jump times ${\delta }_0$:This derivative-jump formula follows by integration by parts on each half-line; the boundary terms equal $-\varphi (0)$ in $G^{\prime \prime }$.

Convolution with the indicator forcing givesThe two values and their first derivatives agree at $\pm 1$, 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 $\varphi$, expand the energy:The mixed term is zero by the weak equation $-u_0^{\prime \prime }+u_0=V$, or integration by parts using the continuous first derivative and decay. The remaining integral is strictly positive whenever $\varphi$ is not identically zero. HenceThe 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.