Absolutely Continuous Spectrum of a Dirac Operator
نویسنده
چکیده
where γj are selfadjoint 4× 4matrices satisfying the relations (2) γjγl + γlγj = 2δl,j . Obviously, if the domain of D0 is defined asH1(R3;C4), then D0 is a selfadjoint operator in L2(R3;C4). The spectrum of D0 is absolutely continuous, it coincides with the complement of the interval (−1, 1) as a set: σ(D0) = σa.c.(D0) = (−∞,−1] ∪ [1,∞). The multiplicity of the spectrum is infinite. We see that R \ σ(D0) = (−1, 1), which means that the interval (−1, 1) is the gap in the spectrum of the operator D0. Let us perturb the operator D0 by an electric potential. One should note that different mathematicians introduce the electric potential in different ways. One of the reasonable ways to introduce it is to add a function V to D0. That means one sets (3) D = D0 + V. Such operators were studied by Birman and Laptev in [2]. Another way of defining DV is to set (4) DV = D0 + V γ0. Such perturbations were, for example, studied by S. Denisov in [6]. Moreover, the free Dirac operator was also introduced in [6] differently. The matrix γ0 in the definition of D0 was omitted. A physicist would say that the Dirac operator in [6] is related to the motion of a massless particle, while the operator (4) is related to the motion of a particle whose mass is equal to 1. Our first result is the following statement.
منابع مشابه
On the Absolutely Continuous Spectrum of Dirac Operator
We prove that the massless Dirac operator in R with long-range potential has an a.c. spectrum which fills the whole real line. The Dirac operators with matrix-valued potentials are considered as well.
متن کاملAbsolutely Continuous Spectrum of Dirac Operators with Square Integrable Potentials
We show that the absolutely continuous part of the spectral function of the one-dimensional Dirac operator on a half-line with a constant mass term and a real, square-integrable potential is strictly increasing throughout the essential spectrum (−∞,−1] ∪ [1,∞). The proof is based on estimates for the transmission coefficient for the full-line scattering problem with a truncated potential and a ...
متن کاملAbsolutely continuous spectrum of Dirac systems with potentials innite at innity
It is shown that the spectrum of a one-dimensional Dirac operator with a potential q tending to infinity at infinity, and such that the positive variation of 1}q is bounded, covers the whole real line and is purely absolutely continuous. An example is given to show that in general, pure absolute continuity is lost if the condition on the positive variation is dropped. The appendix contains a di...
متن کاملSome Observations on Dirac Measure-Preserving Transformations and their Results
Dirac measure is an important measure in many related branches to mathematics. The current paper characterizes measure-preserving transformations between two Dirac measure spaces or a Dirac measure space and a probability measure space. Also, it studies isomorphic Dirac measure spaces, equivalence Dirac measure algebras, and conjugate of Dirac measure spaces. The equivalence classes of a Dirac ...
متن کاملOn the Asymptotics of the Spectral Density of Radial Dirac Operators with Divergent Potential
The radial Dirac operator with a potential tending to infinity at infinity and satisfying a mild regularity condition is known to have a purely absolutely continuous spectrum covering the whole real line. Although having two singular end-points in the limit-point case, the operator has a simple spectrum and a generalised Fourier expansion in terms of a single solution. In the present paper, a s...
متن کامل