MiS Preprint Repository

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MiS Preprint

On Global Attraction to Quantum Stationary States I. Nonlinear Oscillator Coupled to Massive Scalar Field

Alexander Komech and Andrew Komech


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 $t\to\pm\infty$ 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.


Related publications

2007 Repository Open Access
Alexander Komech and Andrew Komech

Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field

In: Archive for rational mechanics and analysis, 185 (2007) 1, pp. 105-142