Comptes Rendus
Mathematical Physics/Functional Analysis
The ground state problem for a quantum Hamiltonian describing friction
Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 151-156.

In this Note, we consider the quantum version of a Hamiltonian model describing friction. This model consists of a particle which interacts with a bosonic reservoir representing a homogeneous medium through which the particle moves. We show that if the particle is confined, then the Hamiltonian admits a ground state if and only if a suitable infrared condition is satisfied. The latter is violated in the case of linear friction, but satisfied when the friction force is proportional to a higher power of the particle speed.

Dans cette Note, on considère la version quantique d'un modèle hamiltonien décrivant le phénomène de frottement. Ce modèle consiste en une particule en interaction avec un réservoir de bosons représentant un milieu homogène dans lequel la particule se déplace. On montre que si la particule est confinée, alors le hamiltonien admet un état fondamental si et seulement si une condition infrarouge adaptée est satisfaite. Cette dernière est violée dans le cas d'un frottement linéaire, mais satisfaite lorsque la force de frottement est proportionnelle à une puissance plus élevée de la vitesse de la particule.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.05.005
Laurent Bruneau 1

1 Department of Mathematical Methods in Physics, Warsaw University, Hoza 74, 00-682, Warszawa, Poland
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Laurent Bruneau. The ground state problem for a quantum Hamiltonian describing friction. Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 151-156. doi : 10.1016/j.crma.2004.05.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.005/

[1] V. Bach; J. Fröhlich; I. Sigal Quantum electrodynamics of confined non-relativistic particles, Adv. Math., Volume 137 (1998), pp. 299-395

[2] L. Bruneau, The ground state problem for a quantum Hamiltonian model describing friction, Preprint mp-arc 03-255, 2003

[3] L. Bruneau; S. De Bièvre A Hamiltonian model for linear friction in a homogeneous medium, Comm. Math. Phys., Volume 229 (2002), pp. 511-542

[4] J. Dereziński; C. Gérard Asymptotic completeness in quantum field theory. Massive Pauli–Fierz Hamiltonians, Rev. Math. Phys., Volume 11 (1999), pp. 383-450

[5] C. Gérard On the existence of ground states for massless Pauli–Fierz Hamiltonians, Ann. Inst. H. Poincaré, Volume 1 (2000), pp. 443-459

[6] J. Glimm; A. Jaffe The λ(ϕ4)2 quantum field theory without cutoffs II. The field operators and the approximate vacuum, Ann. Math., Volume 91 (1970), pp. 362-401

[7] M. Reed; B. Simon Methods of Modern Mathematical Physics, vol. 1, Academic Press, London, 1976

[8] M. Reed; B. Simon Methods of Modern Mathematical Physics, vol. 2, Academic Press, London, 1976

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