Lemma: Slab Green identities

The operators (eq:ER)–(eq:EA) invert $\square +{\mu }^2$ with, respectively, zero initial and zero final Cauchy data. For $F\in {C_c}^{\infty }({M_T}^{\circ })$,

and

These support bounds are compact in the closed slab. Moreover $({{\mathsf{E}}_{\mu }}^{\mathrm{R}}{)}^{\ast }={{\mathsf{E}}_{\mu }}^{\mathrm{A}}$ on interior test functions.

Proof. Differentiating the sine formulas twice verifies the equation and the prescribed endpoint data. Uniqueness follows from the homogeneous Cauchy problem. Applying this uniqueness to ${{\mathsf{E}}_{\mu }}^{\pm }(\square +{\mu }^2)F-F$ gives the left inverse on compact interior tests. Finite propagation gives the support bounds. The intersection of a causal future or past of a compact set with a bounded-time closed slab is compact. Green's identity proves transposition: the retarded solution has zero data at $0$, the advanced solution has zero data at $T$, and there is no spatial boundary contribution. These are the standard smooth Cauchy properties BGP (Chapter 3).