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Journal of Mathematics
Volume 2014, Article ID 713690, 13 pages
http://dx.doi.org/10.1155/2014/713690
Research Article

A Special Class of Infinite Dimensional Dirac Operators on the Abstract Boson-Fermion Fock Space

Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Received 31 May 2014; Accepted 18 August 2014; Published 8 September 2014

Academic Editor: Nasser Saad

Copyright © 2014 Asao Arai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Spectral properties of a special class of infinite dimensional Dirac operators on the abstract boson-fermion Fock space associated with the pair of complex Hilbert spaces are investigated, where is a perturbation parameter (a coupling constant in the context of physics) and the unperturbed operator is taken to be a free infinite dimensional Dirac operator. A variety of the kernel of is shown. It is proved that there are cases where, for all sufficiently large with , has infinitely many nonzero eigenvalues even if has no nonzero eigenvalues. Also Fredholm property of restricted to a subspace of is discussed.