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

Abstract

For $\nu \in [0,1]$ let $D^{\nu }$ be the distinguished self-adjoint realisation of the three-dimensional Coulomb-Dirac operator $-\mathrm{i}\mathbf{\alpha }\cdot \nabla -\nu |\cdot {|}^{-1}$. For $\nu \in [0,1)$ we prove the lower bound of the form $|D^{\nu }|⩾C_{\nu }\sqrt{-\Delta }$, where $C_{\nu }$ is found explicitly and is better then in all previous works on the topic. In the critical case $\nu =1$ we prove that for every $\lambda \in [0,1)$ there exists $K_{\lambda }>0$ such that the estimate $|D^1|⩾K_{\lambda }{a}^{\lambda -1}(-\Delta {)}^{\lambda /2}-a^{-1}$ holds for all $a>0$. As applications we extend the range of coupling constants in the proof of the stability of the relativistic electron-positron field and obtain Cwickel-Lieb-Rozenblum and Lieb-Thirring type estimates on the negative eigenvalues of perturbed projected massless Coulomb-Dirac operators in the Furry picture. We also study the existence of a virtual level at zero for such projected operators.

28 pages, 1 figure