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

For $\nu \in [0,1]$ let $D^{\nu }$ be the distinguished self-adjoint realisation of the three-dimensional Coulomb-Dirac operator $-\mathrm i\boldsymbol\alpha\cdot\nabla -\nu|\cdot|^{-1}$\mathrm{.}For$\nu\in[0, 1)$ we prove the lower bound of the form $\mathrm{\mid }{D}^{\nu }\mathrm{\mid }⩾C_{\nu }\sqrt{-\mathrm{\Delta }}$, 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 $\lambda\in [0, 1)$thereexists$K_\lambda >0$suchthattheestimate$|D^{1}| \geqslant K_\lambda a^{\lambda -1}(-\Delta)^{\lambda/2} -a^{-1}$holdsforall$a >0$. 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.