Atoms confined by very thin layers

TL;DR

This paper analyzes the limiting behavior of an atom confined between two parallel planes as the distance shrinks to zero, showing convergence to a 2D atomic model with Coulomb potential and providing spectral insights.

Contribution

It establishes the norm resolvent convergence of the 3D atomic Hamiltonian to a 2D model as the confinement becomes very thin, justifying the study of 2D atoms with Coulomb interactions.

Findings

Convergence of the Hamiltonian in the limit of zero separation

Reduction to an effective 2D Schrödinger operator

Spectral characterization of the confined atom

Abstract

The Hamiltonian of an atom with $N$ electrons and a fixed nucleus of infinite mass between two parallel planes is considered in the limit when the distance $a$ between the planes tends to zero. We show that this Hamiltonian converges in the norm resolvent sense to a Schr\"{o}dinger operator acting effectively in $L^{2}(\mathbb{R}^{2N})$ whose potential part depends on $a$. Moreover, we prove that after an appropriate regularization this Schr\"{o}dinger operator tends, again in the norm resolvent sense, to the Hamiltonian of a two-dimensional atom (with the three-dimensional Coulomb potential-one over distance), as $a\to 0$. This makes possible to locate the discrete spectrum of the full Hamiltonian once we know the spectrum of the latter one. Our results also provide a mathematical justification for the interest in the two-dimensional atoms with the three-dimensional Coulomb potential.