 Real bounded Fourier modes of the Klein-Gordon equation

With inverse-time convention $e^{i\omega t}$, the half-line Klein-Gordon equation ${\sigma }_{tt}-{\sigma }_{rr}+{\mu }^2\sigma =0$ has spatial normal modes $A{e}^{ikr}+B{e}^{-ikr}$, where $k=\sqrt{{\omega }^2-{\mu }^2}>0$ above threshold and $k=i\sqrt{{\mu }^2-{\omega }^2}$ below it. For the branch even in $\omega$, reality above threshold requires $A(\omega )=B(-\omega {)}^{\ast }$. Below threshold boundedness eliminates $B$, and reality requires $A(\omega )=A(-\omega {)}^{\ast }$. At threshold the spatial differential equation instead has the basis $1,r$; boundedness retains only the constant mode. A spectral integral with ordinary integrable functions as amplitudes ignores the isolated threshold points, while distribution amplitudes can represent the surviving monochromatic constant-spatial modes.