Title: Minimizing the first eigenvalue of Dirac-Coulomb operators: results and conjectures

Speaker: Professor Eric Sere, Université Paris-Dauphine

Beijing time: 16: 00--17: 00 pm, Nov. 12th  (Tuesday), 2024

Paris time: 9: 00--10: 00 am, Nov. 12th  (Tuesday), 2024

Zoom Meeting ID: 626 871 3564, Passcode: 2024

Abstract: This talk is based on joint works with J. Dolbeault, M.J. Esteban and M. Lewin. Consider an electron moving in the attractive Coulomb potential generated by a positive finite measure representing an external charge density. If the total charge is fixed, it is well known that the lowest eigenvalue of the corresponding Schrodinger operator is minimized when the measure is a delta. We investigate the conjecture that the same holds for the relativistic Dirac-Coulomb operator. First we state an abstract result on symmetric operators with gaps. Applied to Dirac-Coulomb operators, this result gives conditions ensuring the existence of a natural self-adjoint realisation and that its eigenvalues are given by min-max formulas Then we define a critical charge such that, if the total charge is fixed below it, then there exists a measure minimising the first eigenvalue of the Dirac-Coulomb operator. We find that this optimal measure concentrates on a compact set of Lebesgue measure zero. The last property is proved using a new unique continuation principle for Dirac operators.

报告人简介：Eric Sere现为法国巴黎九大数学教授，主要从事数学物理、偏微分方程、变分理论等领域的理论研究，特别是在量子多体系统分析领域做出过具有国际影响力的系列研究成果，在Bullet. AMS、CPAM、Duke Math. J.、JEMS等国际顶尖数学期刊上发表论文60余篇，曾担任国际数学杂志AIH. Poincare-Analyse Non Lineaire主编。