# Numerical solutions for a two dimensional quantum dot model

## TL;DR

This paper revisits a 2D quantum dot model, comparing polynomial solutions with new numerical calculations, and extends the spectrum of eigenfunctions and eigenvalues obtainable for the system.

## Contribution

It introduces a more general numerical approach using the Numerov method, enabling calculation of a broader spectrum of eigenfunctions and eigenvalues for the quantum dot model.

## Key findings

- Good qualitative agreement between polynomial and numerical solutions
- Extended the range of external harmonic frequencies for eigenfunction calculation
- Predicted the existence of bound states for the planar system with l=0

## Abstract

In this paper, a quantum dot mathematical model based on a two-dimensional Schr\"odinger equation assuming the 1/r inter-electronic potential is revisited. Generally, it is argued that the solutions of this model obtained by solving a biconfluent Heun equation have some limitations. The known polynomial solutions are confronted with new numerical calculations based on the Numerov method. A good qualitative agreement between them emerges. The numerical method being more general gives rise to new solutions. In particular, we are now able to calculate the quantum dot eigenfunctions for a much larger spectrum of external harmonic frequencies as compared to previous results. Also the existence of bound state for such planar system, in the case l=0, is predicted and its respective eigenvalue is determined.