Properties of Dirac Operators

It has been pointed out before that the self-adjointness of the Dirac operator is a crucial property. Only the self-adjointness guarantees that the time evolution is unitary (which means that the norm is constant in time). Any defect in the self-adjointness of Dirac operators would imply a defect in the unitarity of the time evolution. 

The situation is different with respect to local singularities. Here the Dirac equation is more sensitive than the Schrodinger equation, because the effective angular momentum barrier that could prevent a fall into the singularity is weaker in the relativistic case (as will be discussed in the section about the hydrogen atom below). 

In perturbation theory, one often considers the essential spectrum. It consists of the continuous spectrum together with the accumulation points of the eigenvalues. Whenever the potential vanishes at infinity, the essential spectrum of the Dirac equation is the set 

This is precisely the set of possible energies according to the free Dirac operator. The essential spectrum is obviously rather robust against perturbations. 

The bound-state energies are usually inside the spectral gap (— estimates on the number of bound states have been obtained in [13]. In case of the Coulomb potential there are infinitely many eigenvalues that accumulate at the threshold mcP (or at —mc for a positive Coulomb potential). 

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It should be stressed that the spectral properties of Dirac operators depend very much on the matrix structure of the potential. For a scalar potential V (x) = sc(x) Pj and after a unitary transformation with the constant matrix... 

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