N electrons in a quantum dot: Two-point Pade approximants

TL;DR

This paper develops analytic estimates for the energy levels of 2 to 5 electrons in a 2D quantum dot under a magnetic field using two-point Pade approximants, providing accurate results across different regimes.

Contribution

It introduces a novel application of two-point Pade approximants to estimate energy levels in quantum dots for various electron numbers and magnetic field strengths.

Findings

Approximate energy levels with less than 2.5% error across all regimes.

Derived smooth function of energy as a function of the parameter eta.

Validated the method for N=2 to 5 electrons.

Abstract

We present analytic estimates for the energy levels of N electrons (N = 2 - 5) in a two-dimensional parabolic quantum dot. A magnetic field is applied perpendicularly to the confinement plane. The relevant scaled energy is shown to be a smooth function of the parameter \beta=(effective Rydberg/effective dot energy)^{1/6}. Two-point Pade approximants are obtained from the series expansions of the energy near the oscillator ($\beta\to 0$) and Wigner ($\beta\to\infty$) limits. The approximants are expected to work with an error not greater than 2.5% in the entire interval $0\le\beta < \infty$.