Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

On global attractors and radiation damping for nonrelativistic particle coupled to scalar field


Authors: A. Komech, E. Kopylova and H. Spohn
Original publication: Algebra i Analiz, tom 29 (2017), nomer 2.
Journal: St. Petersburg Math. J. 29 (2018), 249-266
MSC (2010): Primary 35Q60, 78A40, 78M35
DOI: https://doi.org/10.1090/spmj/1492
Published electronically: March 12, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Hamiltonian system of a scalar wave field and a single nonrelativistic particle coupled in a translation invariant manner is considered. The particle is also subject to a confining external potential. The stationary solutions of the system are Coulomb type wave fields centered at those particle positions for which the external force vanishes. It is proved that the solutions of finite energy converge, in suitable local energy seminorms, to the set $ {\mathcal S}$ of all stationary states in the long time limit $ t\to \pm \infty $. Next it is shown that the rate of relaxation to a stable stationary state is determined by the spatial decay of initial data. The convergence is followed by the radiation of the dispersion wave that is a solution of the free wave equation.

Similar relaxation has been proved previously for the case of a relativistic particle when the speed of the particle is less than the wave speed. Now these results are extended to a nonrelativistic particle with velocity, including that greater than the wave speed. However, the research is restricted to the plane particle trajectories in $ \mathbb{R}^3$. Extension to the general case remains an open problem.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35Q60, 78A40, 78M35

Retrieve articles in all journals with MSC (2010): 35Q60, 78A40, 78M35


Additional Information

A. Komech
Affiliation: Faculty of Mathematics, Vienna University, Oskar-Morgenstern-Platz 1, Vienna, Austria; Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia
Email: alexander.komech@univie.ac.at

E. Kopylova
Affiliation: Faculty of Mathematics, Vienna University, Oskar-Morgenstern-Platz 1, Vienna, Austria; Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia
Email: elena.kopylova@univie.ac.at

H. Spohn
Affiliation: Faculty of Mathematics, Technical University of Munich, Boltzmannstraße 3, Garching bei München, Germany
Email: spohn@ma.tum.de

DOI: https://doi.org/10.1090/spmj/1492
Keywords: Hamiltonian system, nonrelativistic particle, wave equation with a source, extended electron
Received by editor(s): November 21, 2016
Published electronically: March 12, 2018
Additional Notes: The research was carried out at the IITP RAS and was supported by the Russian Foundation for Sciences (project no. 14-50-00150)
Dedicated: To the memory of Vladimir Buslaev
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society