The Ground State Problem for a Quantum Hamiltonian Model Describing Friction

Bruneau, Laurent

Canadian journal of mathematics, Tome 59 (2007) no. 5, pp. 897-916

Résumé

In this paper, 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.

Détail

DOI : 10.4153/CJM-2007-038-3

Mots-clés : 81Q10, 46N50

Bibliographie

[A1] Arai, A., On a model of a harmonic oscillator coupled to a quantized, massless, scalar field. I. J. Math. Phys. 22(1981), 2539–2548. Google Scholar

[A2] Arai, A., Ground state of the massless Nelson model without infrared cutoff in a non-Fock representation. Rev.Math. Phys 13(2001), no. 9, 1075–1094. Google Scholar

[AH] Arai, A. and Hirokawa, M., On the existence and uniqueness of ground states of a generalized spin-boson model. J. Func. Anal. 151(1997), no. 2, 455–503. Google Scholar

[BFS1] Bach, V., Fröhlich, J., and Sigal, I., Quantum electrodynamics of confined nonrelativistic particle. Adv. Math. 137(1998), no. 2, 299–395. Google Scholar

[BFS2] Bach, V., Fröhlich, J., and Sigal, I., Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Comm. Math. Phys. 207(1999), no. 2, 249–290. Google Scholar

[B1] Bruneau, L., The ground state problem for a quantum Hamiltonian describing friction. arxivmath-ph/0306011, 2003. Google Scholar

[B2] Bruneau, L., Modèle hamiltonien pour le frottement linéaire en milieu homogène. Ph.D. thesis, Université de Lille-1, 2002. Google Scholar

[BDB] Bruneau, L. and De Bièvre, S., A Hamiltonian model for linear friction in a homogeneous medium. Comm. Math. Phys. 229(2002), no. 3, 511–542. Google Scholar

[D] Dereziński, J., Van Hove Hamiltonians—exactly solvable models of the infrared and ultraviolet problem. Ann. Henri Poincaré, 4(2003), no. 4, 713–738. Google Scholar

[DG1] Dereziński, J. and Gérard, C., Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians. Rev. Math. Phys. 11(1999), no. 4, 383–450. Google Scholar

[DG2] Dereziński, J. and Gérard, C., Scattering theory of infrared divergent Pauli-Fierz Hamiltonians. Ann. Henri Poincaré 5(2004), no. 3, 523–577. Google Scholar

[DJ] Dereziński, J. J. and Jakšić, V., Spectral theory of Pauli-Fierz operators. J. Func. Anal. 180(2001), no. 2, 243–327. Google Scholar

[Fr] Friedrichs, K. O., Mathematical Aspects of Quantum Theory of Fields. Interscience Publishers, New York, 1953. Google Scholar

[G] Gérard, C. C., On the existence of ground states for massless Pauli-Fierz Hamiltonians. Ann. Henri. Poincaré. 1(2000), no. 3, 443–459. Google Scholar

[GJ] Glimm, J. and Jaffe, A., The λ(ϕ4) quantum field theory without cutoffs. II. The field operators and the approximate vacuum. Ann. of Math. 91(1970), 362–401. Google Scholar

[GLL] Griesemer, M., Lieb, E. H., and Loss, M., Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145(2001), no. 3, 557–595. Google Scholar

[KKS] Komech, A., Kunze, M., and Spohn, H., Long-time asymptotics for a classical particle interacting with a scalar wave field. Comm. Partial Differential Equations 22(1997), no. 1-2, 307–335. Google Scholar

[LMS] Lörinczi, J. , Minlos, R. A. and Spohn, H., The infrared behaviour in Nelson's model of a quantum particle coupled to a massless scalar field. Ann. Henri. Poincaré 3(2002), no. 2, 269–295. Google Scholar

[RS1] Reed, M. M. and Simon, B., Methods of Modern Mathematical Physics. 1: Functional Analysis. Academic Press, London, 1976. Google Scholar

[RS2] Reed, M. M. and Simon, B., Methods of Modern Mathematical Physics. 2. Academic Press, London 1976. Google Scholar

[VH] Van Hove, L., Les difficultés de divergences pour un modèle particulier de champ quantifié. Physica 18(1952), 145–159. Google Scholar