Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms

Abstract

Some general results about perturbations of not-semibounded selfadjoint operators by quadratic forms are obtained. These are applied to obtain the distinguished self-adjoint extension for Dirac operators with singular potentials (including potentials dominated by the Coulomb potential with Z<137). The distinguished self-adjoint extension, is the unique selfadjoint extension, for which the wave functions in its domain possess finite mean kinetic energy. It is shown moreover that the essential spectrum of the distinguished extension is contained in the spectrum of the free Hamiltonian.

Spectral boundary-value problems for the Dirac system with a singular potential

“…Now we briefly discuss the abstract approach proposed in [9]. As has already been mentioned, it was used in [9] for the investigation of the Dirac operator with singular potential in R 3 , and the result cited above was obtained in [9] (see Proposition 2.1). In the next section we show that Nenciu's approach enables us to define the operator (D + V ) Ω in an alternative way.…”

“…The Dirac operator with singular potential in R 3 . The Dirac operator in R 3 with a potential that has a Coulomb singularity of type (2.1) was investigated among others by Nenciu [9], Wüst [20]- [22], Klaus-Wüst [23,24], and Schmincke [25]. In [9] quadratic and sesquilinear forms were used, so the approach is similar to the classical approach to boundary-value problems for strongly elliptic second order systems.…”

“…Furthermore, assume that the operator This assertion is contained in [9, Corollary 2.1]. In [9] that corollary was applied in R 3 to the free Dirac operator as A, multiplication by the singular part of the potential as V , multiplication by its smooth part as W , and the full Dirac operator with the potential consisting of the singular and smooth parts as B. In the next section we shall show that our operator (D + V ) Ω can be obtained in a similar way if for A we take either the operator (D) Ω corresponding to the free Dirac system in Ω (if this operator has bounded inverse) or, otherwise, the operator (D + c) Ω with a small real constant c added to gain invertibility.…”

“…The operator (D + V ) Ω acts boundedly from this subspace to L 2 (Ω) 4 . The spectrum of this operator is discrete, there exists an orthonormal basis of eigenfunctions in L 2 (Ω) 4 , and the eigenvalues have the usual asymptotic behavior (9) λ n = bn 1/3 + O(1) (n → ±∞), b = 3π 2 meas Ω 1/3 .…”

“…We overcome this difficulty by considering the difference of resolvents for the perturbed and nonperturbed operators. Here the results by Nenciu [9] are essentially used, namely, the abstract approach for nonsemibounded operators with the study of the resolvent, and realization of this approach in R 3 . First we show that the operator we define for our problem is also a realization of the abstract Nenciu approach.…”