Liouville phenomenon for the Klein–Gordon equation

The Klein–Gordon equation is the simplest hyperbolic PDE after the wave equation itself. It models the wave function of a free boson with positive rest mass. We carry out our analysis in one spatial and one temporal dimension only. The bounded solutions to the Klein–Gordon equation exhibit a strong relation between its values along two characteristic axes. However, this is not expected from studying e.g. the wave equation. Here, we explore the bounds (not by constants) that force such a relationship.