Moscow Mathematical Society

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(s): 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
Posted: November 19, 2008
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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.

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R. A. Minlos
Affiliation: Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia
Email: minl@iitp.ru

DOI: 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
Posted: 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.
Copyright of article: Copyright 2008, American Mathematical Society

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