Exact local construction

The independent probe is another real massless scalar $\psi$. Both fields initially have the same free operator $P$. The local linear system–probe framework supplies the coupled Green operators and Weyl scattering interpretation FewsterVerch,FewsterJubbRuep. We now give the particular construction and estimates.

Fix $L,T$ and abbreviate $B=B_L,b=b_L$. Choose a spacelike-compact free probe solution $v_{\mathrm{bulk}}$ equal to one near $\mathrm{supp}B$. To construct it, prescribe compact Cauchy data at $t_{\mathrm{m}}-T-1$ equal to the constant solution and its zero velocity on the causal predecessor of $\mathrm{supp}B$. Finite propagation and uniqueness give the desired equality.

Write $v_{\mathrm{bulk}}=Y_{0}v/r$. Choose a future switch ${\eta }_{+}$, zero before $t_{\mathrm{m}}+T+1$ and one after $t_{\mathrm{m}}+T+2$, and set

Then

Couple the fields by

with coupled operator

The final bounded probe effect is

Let