{"paper":{"title":"Lower bounds on the moduli of three-dimensional Coulomb-Dirac operators via fractional Laplacians with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"David M\\\"uller, Sergey Morozov","submitted_at":"2016-12-20T10:26:54Z","abstract_excerpt":"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 applic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.06591","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}