Note on the numerical solution of the scalar Helmholtz equation in a nanotorus with uniform Dirichlet boundary conditions

TL;DR

This paper explores the numerical solution of the scalar Helmholtz equation within a nanotorus, analyzing eigenfunctions and eigenvalues, and drawing parallels with related physical systems to classify eigenfunctions.

Contribution

It introduces a classification scheme for eigenfunctions of the Helmholtz equation in a nanotorus based on symmetry and indices, expanding understanding of their properties.

Findings

Eigenvalues and eigenfunctions are characterized for the nanotorus

Symmetry properties of eigenfunctions are discussed

A classification scheme based on three indices is proposed

Abstract

This note describes the solution of the Helmholtz equation inside a nanotorus with uniform Dirichlet boundary conditions. The eigenfunction symmetry is discussed and the lower-order eigenvalues and eigenfunctions are shown. The similarity with the case of a long cylinder and with that of the vibrations of a circular elastic membrane is discussed. This similarity is used to propose a classification scheme of the eigenfunctions based on three indices.