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

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. 1. Introduction In a previous paper [1] (cf. [2]), the author introduced a general class of infinite dimensional Dirac type operators on the abstract boson-fermion Fock space associated with the pair of complex Hilbert spaces (for the definition of and , see Section 2), where is a densely defined closed linear operator from to . The operator gives an infinite dimensional and abstract version of finite dimensional Dirac type operators. In applications to physics, unifies self-adjoint supercharges (generators of supersymmetry) of some supersymmetric quantum field models (e.g., [3–6]). In the paper [1], basic properties of are discussed. As for the spectral properties of , only the zero-eigenvalue of is considered. Moreover, a class of perturbations for is introduced and, under a suitable condition, a path (functional) integral representation for the index of the perturbed operator restricted to a subspace of is established. The (essential) self-adjointness of is partially discussed in [7]. Analysis of other properties of including spectral ones except for the zero-eigenvalue has been left open. In this paper, we undertake a comprehensive operator-theoretical analysis for . But, as a preliminary, we consider, in the present paper, only the case where is a simple form and see what kind of phenomena occurs under the perturbation . The outline of the present paper is as follows. In Section 2 we review briefly some contents in [1]. In Section 3 we identify the spectra of . This is done via spectral analysis of a second quantization operator on the boson-fermion Fock space . In Section 4, we introduce a perturbation with and , where is a perturbation parameter (a coupling constant in the context of physics). The perturbed operator is the main object of our analysis in the present paper. We prove the self-adjointness of and the essential self-adjointness of it on a suitable dense subspace of with some other properties (Theorem 14). Also the spectra of are