On the Virtual Levels of Positively Projected Massless Coulomb–Dirac Operators

Abstract

Considering different self-adjoint realisations of positively projected massless Coulomb-Dirac operators we find out, under which conditions any negative perturbation, however small, leads to emergence of negative spectrum. We also prove some weighted Lieb-Thirring estimates for negative eigenvalues of such operators. In the process we find explicit spectral representations for all self-adjoint realisations of massless Coulomb-Dirac operators on the half-line.

Lieb–Thirring and Cwickel–Lieb–Rozenblum inequalities for perturbed graphene with a Coulomb impurity

“…Theorem 4 is a form of Lieb-Thirring inequality, (see [20] for the original result and [16] for a review of further developments). In another publication [21] we prove that D 1/2 (0, V ) has a negative eigenvalue for any non-trivial V 0. This situation is associated with the existence of a virtual level at zero, as observed for example for the operator − d 2 dr 2 − 1 4r 2 in L 2 (R + ) (see [9], Proposition 3.2).…”

Lower bounds on the moduli of three-dimensional Coulomb-Dirac operators via fractional Laplacians with applications

“…The proof is analogous to the one of Part I of Theorem 2.5 in [13]. Let j ∈ {−1, 1} 2 be such that the conditions (1.11) and (1.12) are satisfied.…”