 Solution

At fixed frequency, the transformed equation has characteristic roots $\pm ik$, whereFor a consistent explicit branch chooseThis branch is even in $\omega$. Away from the two thresholds, the spatial solution is $\tilde{\sigma }=A(\omega )e^{ikr}+B(\omega )e^{-ikr}$, so Fourier inversion givesAt $|\omega |=\mu$, the spatial equation has a double zero characteristic root and its full solution is $C+Dr$. The two exponentials at $k=0$ represent only its constant part. This isolated-frequency issue does not change an integral with ordinary integrable amplitudes; threshold-supported distributional modes require separate treatment.