Sign and measure checks

The scalar spectral equation is

Therefore

and

Using $+{{\int }_t}^{\infty }{W}_B(t-s)f(s)\mathrm{d}s$ would instead give the negative of the required advanced inverse.

The kernels are normalized relative to $\mathrm{d}t\mathrm{d}V=c\mathrm{d}T\mathrm{d}V$. If one instead writes the source integral using $\mathrm{d}T\mathrm{d}V$, the corresponding kernel is

No additional factor is needed when the host volume density $c\mathrm{d}T\mathrm{d}V$ is used.

The transverse spectral statement ${\mathrm{spec}}_{\mathrm{ess}}(B_{\perp })=[0,\infty )$ must not be substituted for the full product statement (eq:full-spectrum). A transverse bound state times a plane wave in $z$ is not an $L^2(N)$ eigenfunction.