= Solution
Reality requires $\widetilde\sigma(\omega,r)=\widetilde\sigma(-\omega,r)^*$. The way it constrains the amplitudes depends on whether the spatial <wave number> is real or imaginary.
For $|\omega|>\mu$, $k$ is real and even. <Complex conjugation> interchanges the two exponentials, so comparing independent spatial factors gives
$$
\boxed{A(\omega)=B(-\omega)^*\qquad(|\omega|>\mu).}
$$
The companion equation $B(\omega)=A(-\omega)^*$ is the same constraint with <frequency> reversed and conjugated. Neither oscillatory spatial branch grows as $r\to\infty$.
For $|\omega|<\mu$, put $k=i\kappa$ with $\kappa>0$. Then
$$
\widetilde\sigma=A(\omega)e^{-\kappa r}+B(\omega)e^{\kappa r}.
$$
Boundedness excludes the growing exponential, and reality then pairs the remaining amplitudes:
$$
\boxed{B(\omega)=0,\qquad A(\omega)=A(-\omega)^*\qquad(|\omega|<\mu).}
$$
These are the two independent low-frequency requirements. At threshold boundedness eliminates the separate linear-in-$r$ solution discussed in (ii). The constraints are understood for a <Fourier transform> representation in which spatial mode amplitudes are well defined; sufficient amplitude integrability also ensures the spectral <integral> itself is bounded.
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