Atoms confined by very thin layers

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.