Consider the partial differential equation:
$$\left(\Delta-\frac{\partial^2}{\partial t^2}\right)u(x,t)=V(x)u(x,t),\qquad(\bigstar)$$
for $x\in\mathbb{R}$, $t\in\mathbb{R}$, $V\in C^\infty_0(\mathbb{R})$.
One may determine that
$$u(x,t)=\delta(t+x-s)+K(x,s-t)\qquad(\bigstar\bigstar)$$
is a solution to $(\bigstar)$, where $K$ is the "dual fundamental solution" of $(\bigstar)$.
What does "dual fundamental solution" even mean?
Note that I am familiar with the notion of fundamental solutions which are distributional quantities, $E$, satisfying $\mathscr{L}E=\delta,$ where $\mathscr{L}$ is a partial differential operator.
If we multiply $(\bigstar\bigstar)$ by $D(s)$, where $D:\mathbb{R}\to\mathbb{R}$, the scattered data, is the reflection of the plane wave from the potential $V$, then we obtain
$$D(s)\delta(t+x-s)+D(s)K(x,s-t)\qquad(\bigstar\bigstar\bigstar)$$
Then integrating $(\bigstar\bigstar\bigstar)$ in $s$, we get
$$\int_\mathbb{R}D(s)\delta(t+x-s)\,\mathrm{d}s+\int_\mathbb{R}D(s)K(x,s-t)\,\mathrm{d}s$$
Then, from this, how do we see that $$D(t+x)+\int_\mathbb{R}D(s+t)K(x,s)\,\mathrm{d}s$$ is also a solution to $(\bigstar)$?