The lower part of the spectrum of the Hamiltonian of the spinless Pauli-Fierz model (A two-component Bose field interacting with a charged particle)

Author:
R. A. Minlos

Translated by:
E. Khukhro

Original publication:
Trudy Moskovskogo Matematicheskogo Obshchestva, tom **69** (2008).

Journal:
Trans. Moscow Math. Soc. **2008**, 209-253

MSC (2000):
Primary 81Q10; Secondary 47A10, 47A40, 47A55, 81T10, 81U99, 81V10

Published electronically:
November 19, 2008

MathSciNet review:
2549448

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate a model of a vector massive spinless Bose field in the space interacting with a non-relativistic particle, where the interaction parameter is assumed to be sufficiently small. We study the ground state of the Hamiltonian for a fixed total momentum of the system and show that such a state is non-degenerate and exists only for a bounded domain of values of this momentum. We also show that, apart from the ground state, the operator has no other eigenvalues below the continuous spectrum. Furthermore, the next two, ``one-boson'', branches of the spectrum of are constructed, which describe the scattering of one boson (with two possible polarization values) on the ground state.

**1.**Nicolae Angelescu, Robert A. Minlos, and Valentin A. Zagrebnov,*Lower spectral branches of a particle coupled to a Bose field*, Rev. Math. Phys.**17**(2005), no. 10, 1111–1142. MR**2187291**, 10.1142/S0129055X05002509**2.**Asao Arai,*Spectral analysis of a quantum harmonic oscillator coupled to infinitely many scalar bosons*, J. Math. Anal. Appl.**140**(1989), no. 1, 270–288. MR**997857**, 10.1016/0022-247X(89)90108-X**3.**V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko,*Singularities of differentiable maps. Vol. II*, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. Monodromy and asymptotics of integrals; Translated from the Russian by Hugh Porteous; Translation revised by the authors and James Montaldi. MR**966191****4.**Volker Bach, Jürg Fröhlich, and Alessandro Pizzo,*Infrared-finite algorithms in QED: the groundstate of an atom interacting with the quantized radiation field*, Comm. Math. Phys.**264**(2006), no. 1, 145–165. MR**2212219**, 10.1007/s00220-005-1478-3**5.**V. Bach, J. Fröhlich, and I.M. Sigal,- 1.
Volker
Bach, Jürg
Fröhlich, and Israel
Michael Sigal,
*Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field*, Comm. Math. Phys.**207**(1999), no. 2, 249–290. MR**1724854**, 10.1007/s002200050726 - 2.
Volker
Bach, Jürg
Fröhlich, and Israel
Michael Sigal,
*Quantum electrodynamics of confined nonrelativistic particles*, Adv. Math.**137**(1998), no. 2, 299–395. MR**1639713**, 10.1006/aima.1998.1734

Retrieve articles in *Transactions of the Moscow Mathematical Society*
with MSC (2000):
81Q10,
47A10,
47A40,
47A55,
81T10,
81U99,
81V10

Retrieve articles in all journals with MSC (2000): 81Q10, 47A10, 47A40, 47A55, 81T10, 81U99, 81V10

Additional Information

**R. A. Minlos**

Affiliation:
Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

Email:
minl@iitp.ru

DOI:
http://dx.doi.org/10.1090/S0077-1554-08-00170-2

Keywords:
Non-relativistic charged particle,
Bose field,
Pauli--Fierz model,
Hamiltonian,
spectrum,
ground state,
scattering

Published electronically:
November 19, 2008

Additional Notes:
The author thanks Professor H. Spohn and Dr. E. A. Zhizhina for useful discussions on questions relating to this paper. The author thanks the Mathematics Centre of the Munich Technical University, where the plan for this research emerged for the first time, for its warm hospitality and financial support. The author also thanks the following organisations for financial support: the Russian Foundation for Basic Research (grant #05-01-00449), the President Foundation for Support of Scientific Schools of Russia, and the American Foundation CRDF, grant RUM1-2603-MO-05.

Article copyright:
© Copyright 2008
American Mathematical Society