A Polynomial Approach to the Spectrum of Dirac-Weyl Polygonal Billiards

TL;DR

This paper introduces a variational method to approximate the low-energy spectrum and wave-functions of Dirac-Weyl particles in arbitrary convex polygonal quantum billiards, extending techniques from vibrational mode analysis.

Contribution

It develops a novel variational approach for the Dirac-Weyl equation in polygonal enclosures, applicable to quantum dots with massless electrons, and demonstrates its convergence and effectiveness.

Findings

Method converges for known spectra

Accurately approximates unknown spectra

Applicable to various boundary conditions

Abstract

The Schr\"odinger equation in a square or rectangle with hard walls is solved in every introductory quantum mechanics course. Solutions for other polygonal enclosures only exist in a very restricted class of polygons, and are all based on a result obtained by Lam\'e in 1852. Any enclosure can, of course, be addressed by finite element methods for partial differential equations. In this paper, we present a variational method to approximate the low-energy spectrum and wave-functions for arbitrary convex polygonal enclosures, developed initially for the study of vibrational modes of plates. In view of the recent interest in the spectrum of quantum dots of two dimensional materials, described by effective models with massless electrons, we extend the method to the Dirac-Weyl equation for a spin-1/2 fermion confined in a quantum billiard of polygonal shape, with different types of boundary conditions. We illustrate the method's convergence in cases where the spectrum in known exactly and apply it to cases where no exact solution exists.