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

Abstract

We study the two dimensional massless Coulomb-Dirac operator restricted to its positive spectral subspace and prove estimates on the negative eigenvalues created by electromagnetic perturbations. (2010): 35P15, 35Q40. Mathematics Subject Classification

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

“…Such operators are relevant for description of graphene with Coulomb impurity [11]. Every u ∈ L 2 (R 2 ) can be represented in the polar coordinates as Introducing the unitary angular momentum decomposition…”

“…such that the self-adjoint operator in question coincides with 11) where the components on the right hand side are defined in Theorem 1. On the other hand, every θ satisfying (2.10) gives rise to a self-adjoint realisation…”

“…For ν ∈ [0, 1/2) an application of Theorem 5, part 2 to D ν (Q) gives a weaker result than that of Theorem 3 in [11], which gives the following bound on the amount of negative eigenvalues of…”

“…The case of ν = 0 is trivial. Otherwise we proceed as in the proof of Lemma 28 in [11]. The only difference is that we now have relaxed the assumptions on κ and ν. holds for all κ satisfying (7.1) and s ∈ R. Since (7.3) does not depend on the signs of κ and s, without loss of generality we can assume κ, s ∈ R + .…”

“…In a related paper [11] the authors have obtained estimates of Cwikel-LiebRozenblum and Lieb-Thirring types on the negative eigenvalues of the perturbed positively projected two-dimensional massless Coulomb-Dirac operator emerging in the study of graphene. For the critical value of the coupling constant the control on the number of eigenvalues failed, which naturally posed a question about the existence of a virtual level at zero, i.e.…”