Asymptotics for Two-dimensional Atoms

2011/02/21 by Phan Thanh Nam, Phan Thành Nam, Nam, Phan Thanh +4
Mathematics · Physics and Astronomy · #81Q20 #81V45 #Advanced Chemical Physics Studies #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP #msc:81Q20 #msc:81V45

paper · pdf · doi:10.48550/arxiv.1102.4229

Revised version to appear in Ann. Henri Poincaré

openalex publication_date 2011/02/21 · arxiv created 2011/06/16 · arxiv updated 2011/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the ground state energy of an atom confined to two dimensions with an infinitely heavy nucleus of charge Z>0 and N quantum electrons of charge -1 is E(N,Z)=-1/2Z²ln Z+(E^\TF(λ)+1/2c^{\rm H})Z²+o(Z²) when Z→ ∞ and N/Z→ λ, where E^\TF(λ) is given by a Thomas-Fermi type variational problem and c^{\rm H}≈ -2.2339 is an explicit constant. We also show that the radius of a two-dimensional neutral atom is unbounded when Z→ ∞, which is contrary to the expected behavior of three-dimensional atoms.

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