Quantum mechanics

Ecology

Pure mathematics

Mathematical physics

Biology

Schrödinger's cat

Type (biology)

Physics

Nonlinear system

Mathematics

Pullback attractor

Invariant (physics)

Attractor

Mathematical analysis

Klein–Gordon equation

Journal of Differential Equations

In this article, the authors investigate the system of Schrödinger and Klein-Gordon equations with Yukawa coupling. They first prove the existence of pullback attractor and construct a family of invariant Borel probability measures. Then they establish that this family of probability measures satisfies a Liouville type theorem and is indeed a statistical solution for the coupling equations. Further, they reveal that the invariant property of the statistical solution is a particular situation of the Liouville type theorem.

Statistical solution and Liouville type theorem for the Klein-Gordon-Schrödinger equations