MiS Preprint Repository

The long-time asymptotics is analyzed for all finite energy solutions to a model $\mathbf{U}(1)$-invariant nonlinear Klein-Gordon equation in one dimension, with the nonlinearity concentrated at a point. Our main result is that each finite energy solution converges as $t\to \pm \mathrm{\infty }$ to the set of "nonlinear eigenfunctions" $\psi \pm (x)e^{-i\omega \pm t}$.

Let us name the main steps, which also constitute the novelties of our approach:
a) We analyze the time-spectrum of the solution to the nonlinear wave equation by the Fourier-Laplace transform;
b) We establish the absolute continuity of the spectral density outside the spectral gap;
c) We establish compactness of the spectral density inside the spectral gap in the class of quasimeasures;
d) We reduce any omega-limiting spectral density to a delta-function applying the classical Titchmarsh theorem of Harmonic Analysis.

The research is inspired by Bohr's postulate on quantum transitions and Schrödinger's identification of the quantum stationary states to the eigenfunctions of the coupled $\mathbf{U}(1)$-invariant Maxwell-Schrödinger or Maxwell-Dirac equations.