Proposition: Witness source support and noncompactness

For $\lambda >0$, $S_{\lambda }$ is smooth, vanishes for $t\le 0$, is flat at $0$, and is past compact in the sense of Definition def:past. Nevertheless,

At $\lambda =0$, the displayed source is zero.

Proof. Smoothness, past vanishing, and flatness follow from (eq:switch), (eq:Slambda), and (eq:d1). Lemma lem:past proves past compactness.

For $\epsilon <t<T$,

This is nonzero at every spatial point. Thus the spatial support is unbounded and the source is not compactly supported in the open slab.

Every term of (eq:Slambda) has a factor $\lambda$ or ${\lambda }^2$, proving the assertion at zero frequency.