Bound states in two-dimensional shielded potentials

TL;DR

This paper investigates the existence of bound states in two-dimensional shielded potentials, revealing that such states can occur even when traditional conditions are not met, with implications for surface impurity experiments.

Contribution

It demonstrates that bound states can exist in 2D shielded potentials beyond classical criteria and explores electron-electron interactions leading to bound states at specific densities.

Findings

Bound states can exist in shielded potentials with positive integral conditions.

Electron-electron interactions can produce bound states at certain densities.

Numerical analysis confirms the theoretical predictions.

Abstract

We study numerically the existence and character of bound states for positive and negative point charges shielded by the response of a two-dimensional homogeneous electron gas. The problem is related to many physical situations and has recently arisen in experiments for impurities on metal surfaces with Schockley surface states. Mathematical theorems ascertain a bound state for two-dimensional circularly symmetric potentials $V(r)$ with $\int_{0}^{\infty}{dr r V(r)} \leq 0$. We find that a shielded potential with $\int_{0}^{\infty}{dr r V(r)} > 0$ may also sustain a bound state. Moreover, on the same footing we study the electron-electron interactions in the two-dimensional electron gas finding a bound state with an energy minimum for a certain electron gas density.