Liouville phenomenon for the Klein-Gordon equation in 1+1 dimensions

TL;DR

This paper investigates the Klein-Gordon equation in 1+1 dimensions, revealing a Liouville phenomenon in spacelike regions where solutions exhibit symmetry due to growth constraints, linking to classical harmonic function principles.

Contribution

It demonstrates the Liouville phenomenon for solutions of the Klein-Gordon equation in spacelike quarter-planes, highlighting new symmetry properties based on growth conditions.

Findings

Solutions in spacelike regions show symmetry due to growth constraints

The Liouville phenomenon is analogous to classical harmonic function principles

Different behaviors are observed in timelike versus spacelike regions

Abstract

We study the Klein-Gordon equation in one spatial and one temporal dimension. Physically, this equation describes the wave function of a relativistic spinless boson with positive rest mass. Mathematically, this is the most elementary hyperbolic partial differential equation, after the wave equation itself. Relative to the origin, the spacetime splits according to the light cones, and we find four quarter-planes, two of which are timelike while the remaining two are spacelike. Not unexpectedly, the solutions behave quite differently in the two types of quarter-planes. It turns out that the spacelike quarter-planes exhibit a Liouville phenomenon, where insufficient growth forces the solutions to display a certain kind of symmetry, where the values on the two linear edges are in a one-to-one relation. This phenomenon shares features with the classical Liouville theorem as well as the Phragmen-Lindelof principle for harmonic functions.