Let $\mathcal{L}$ be the operator

$$\mathcal{L} y=\frac{d^{2} y}{d x^{2}}-k^{2} y$$

on functions $y(x)$ satisfying $\lim _{x \rightarrow-\infty} \quad y(x)=0$ and $\lim _{x \rightarrow \infty} y(x)=0$.

Given that the Green's function $G(x ; \xi)$ for $\mathcal{L}$ satisfies

$$\mathcal{L} G=\delta(x-\xi)$$

show that a solution of

$$\mathcal{L} y=S(x)$$

for a given function $S(x)$, is given by

$$y(x)=\int_{-\infty}^{\infty} G(x ; \xi) S(\xi) d \xi$$

Indicate why this solution is unique.

Show further that the Green's function is given by

$$G(x ; \xi)=-\frac{1}{2|k|} \exp (-|k||x-\xi|)$$