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Liouville phenomenon for the Klein-Gordon equation

Haakan Hedenmalm

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.

Liouville phenomenon for the Klein-Gordon equation

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.
Paper Structure (27 sections, 20 theorems, 191 equations)

This paper contains 27 sections, 20 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Classical backgrund
  3. Finite speed propagation and uniqueness
  4. Poincaré group invariance of the Klein-Gordon equation
  5. Gluing solutions of the Klein-Gordon equation along characteristics
  6. Splitting Riemann's solution into unilateral wave solutions
  7. Motivation: the Phragmén-Lindelöf principle for harmonic functions
  8. Liouville type phenomenon for spacelike quarter-planes
  9. Acknowledgements
  10. Horizontal unilateral wave solutions
  11. An example of growth in a spacelike quadrant
  12. The evolution equation for horizontal unilateral wave solutions
  13. Uniqueness in the exponential growth case $q=1$
  14. An auxiliary integral estimate
  15. Estimation of an integral
  16. ...and 12 more sections

Key Result

Theorem 1.2

(Riemann) Provided that $f,g$ are continuous univariate functions with $f(0)=g(0)$, the function $u(x,y)=\mathbf{R}[f,g](x,y)$ is a continuous function of two variables, which solves the Darboux-Goursat problem eq:KG+Goursat-1 in the sense of distribution theory.

Theorems & Definitions (46)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Definition 1.9
  • Proposition 1.10
  • ...and 36 more