A Hardy-type inequality and some spectral characterizations for the Dirac–Coulomb operator

Abstract

We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrixvalued potentials V of Coulomb type: we characterise its eigenvalues in terms of the Birman-Schwinger principle and we bound its discrete spectrum from below, showing that the groundstate energy is reached if and only if V verifies some rigidity conditions. In the particular case of an electrostatic potential, these imply that V is the Coulomb potential.

Dirac–Coulomb operators with general charge distribution II. The lowest eigenvalue

“…The formula ( 22) can be interpreted in the form of Hardy-type inequalities similar to those studied in [4,8,[12][13][14]]. Indeed, denoting again V μ = μ * |x| −1 for shortness, the first line in (22)…”

“…in the Hilbert space L 2 (R 3 , C 4 ), where α 1 , α 2 , α 3 , β are the Dirac matrices recalled below in (8). One important difference with the Schrödinger case is that the spectrum of D 0 is R \ (−1, 1), hence is unbounded both from below and above.…”

Boundary triples for the Dirac operator with Coulomb-type spherically symmetric perturbations

“…It is standard (see for example [8,Lemma 2.1]) to show that the last term in the previous equation vanishes, indeed 1 + 2S · L and ∂ r + 1 |x| are respectively symmetric and skew-symmetric on C ∞ c (R 3 ) 4 , and the two operators commute with each other. Let φ := |x|ψ.…”

Keller-type bounds for Dirac operators perturbed by rigid potentials

“…The combination of the Birman-Schwinger principle with resolvent estimates for free operators is one of the way to approach the localization problem for eigenvalues: it has been widely employed in the later times (see e.g. [26,12,19,30,10,25,23,4,5,18,17] among others) and it will be the approach we are going to follow in this work too, as we will see. However, despite the robustness of the Birman-Schwinger principle, it is not the only tool one could use to obtain spectral enclosures for non-selfadjoint operators: another powerful technique is the method of multipliers, see e.g.…”