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arXiv · 1612.06591

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

Sergey Morozov·David Müller
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Abstract

For $ν\in[0, 1]$ let $D^ν$ be the distinguished self-adjoint realisation of the three-dimensional Coulomb-Dirac operator $-\mathrm i\boldsymbolα\cdot\nabla -ν|\cdot|^{-1}$. For $ν\in[0, 1)$ we prove the lower bound of the form $|D^ν| \geqslant C_ν\sqrt{-Δ}$, where $C_ν$ is found explicitly and is better then in all previous works on the topic. In the critical case $ν=1$ we prove that for every $λ\in [0, 1)$ there exists $K_λ>0$ such that the estimate $|D^{1}| \geqslant K_λa^{λ-1}(-Δ)^{λ/2} -a^{-1}$ 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.

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Sergey Morozov, David Müller. 2016-12-20. Lower bounds on the moduli of three-dimensional Coulomb-Dirac operators via fractional Laplacians with applications. https://doi.org/10.1063/1.4995406

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