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

TL;DR

This paper establishes new lower bounds for three-dimensional Coulomb-Dirac operators using fractional Laplacians, improving previous results and applying these bounds to stability analysis and eigenvalue estimates in relativistic quantum models.

Contribution

It provides explicit lower bounds for Coulomb-Dirac operators in 3D, including the critical case, and applies these bounds to stability and spectral estimates in quantum field models.

Findings

Derived explicit bounds for |D^ν| in terms of fractional Laplacians.

Extended the stability range of the relativistic electron-positron field.

Obtained eigenvalue estimates and studied virtual levels at zero.

Abstract

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