Asymptotics for Two-Dimensional Atoms

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 −1when Z → ∞ and N/Z → λ, where E TF (λ) is given by a Thomas-Fermi type variational problem and c 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 threedimensional atoms.

A Generalization of Weyl's Asymptotic Formula for the Relative Trace of Singular Potentials

“…Furthermore, the method generalizes to all d ≥ 2; only the parameter adjustment is different. Our method generalizes the proof in [5].…”

“…In the two-dimensional case, the Coulomb potential −|x| −1 is singular in the sense that it is not in L 2 loc . See Nam, Portmann and Solovej [5], where the authors prove this result in d = 2 for potentials with Coulomb-like singularities.…”

On the Number of Electrons That a Nucleus Can Bind

“…Although this bound was already proved in the Hartree-Fock theory [32], in Schrödinger's theory it is only known that R Z ≥ CZ −5/21 [25]. On the other hand, for atoms restricted to two dimensions, we have R Z → ∞ as Z → ∞ [23]. 4.…”

A New Estimate on the Two-Dimensional Indirect Coulomb Energy

“…Recently, Benguria, Bley, and Loss obtained an alternative to (3), which has a lower constant (close to 1.45) to the expense of adding a gradient term (see Theorem 1.1 in [2]). After the work of Lieb and Oxford [13] many people have considered bounds on the indirect Coulomb energy in lower dimensions (in particular see, e.g., [9] for the one dimensional case, [16], [21], [23] and [24] for the two dimensional case, which is important for the study of quantum dots). In this manuscript we give an alternative to the Lieb-Solovej-Yngvason bound [16], with a constant much closer to the numerical values proposed in [24] (see also the references therein) to the expense of adding a gradient term.…”