On the spectrum of the perturbed Dirac operator
by
P.A. Cojuhari
Moldova State University

The free Dirac operator is given (see, for instance, , , ), in the Hilbert space L²(Rⁿ)\otimesC^m, by

H0=-i \alpha·Ñ+\alpha0  ,

where \alpha·Ñ = \sum_k=1ⁿ\alpha_k\frac\partial\partialx_k,  \alpha_k  (k=0, 1, ..., n) being as Hermitian m×m matrices satisfy the Clifford relations: \alpha_j\alpha_k+\alpha_k\alpha_j = 2\delta_jk  (j, k=0, 1, ..., n). The matrices \alpha_k  (k=0, 1, ..., n) can be taken to belong to GL(m;C) with m=2^\fracn2 for n even and m=2^\fracn+12 for n odd.

We pertub the operator H₀ with a multiplicative operator Q:

(Qu)(x)=Q(x)u(x)     (x in Rn; u in L2(Rn)\otimesCm) ,

where Q(x)=[q_jk(x)]₁^m,  x in Rⁿ, is a Hermitian matrix-valued function with q_jk (j, k=1, ..., m) from the space L_\infty(Rⁿ). Denote H=H₀+Q. It is assumed that the operators H₀ and H are defined on the same domain H¹(Rⁿ)\otimesC^m.

Our purpose is to study the spectral properties of the perturbed Dirac operator H. In particular, a discussion on the structure of the spectrum of the operator H is undertaken.

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V.A.Fock, The Origine of the Quantum Mechanics, [Russian], Nauka, Moscow, 1972.

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I.M.Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators , [Russian], Fizmatgiz, Moscow, 1963.

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B.Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, Berlin-Heidelberg-New York, 1992.

Date received: May 29, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeo-34.