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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

 

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
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Abstract | References | Similar Articles | Additional Information

Abstract: We investigate a model of a vector massive spinless Bose field in the space $ R^3$ interacting with a non-relativistic particle, where the interaction parameter is assumed to be sufficiently small. We study the ground state of the Hamiltonian $ H_P$ for a fixed total momentum $ P$ 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 $ H_P$ has no other eigenvalues below the continuous spectrum. Furthermore, the next two, ``one-boson'', branches of the spectrum of $ H_P$ are constructed, which describe the scattering of one boson (with two possible polarization values) on the ground state.


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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
PII: S 0077-1554(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