An extension of Fourier analysis for the n-torus in the magnetic field and its application to spectral analysis of the magnetic Laplacian

TL;DR

This paper extends Fourier analysis techniques to the n-dimensional torus in a magnetic field, providing explicit solutions to the magnetic Laplacian eigenvalue problem and characterizing the quantum mechanics in this setting.

Contribution

It introduces a new Fourier analysis method for magnetic tori and defines the magnetic algebra, fully characterizing the Laplacian's spectral properties in this context.

Findings

Explicit eigenfunctions of the magnetic Laplacian are derived.

The magnetic algebra's irreducible representations describe the function space.

A complete set of solutions for the Schrödinger equation in magnetic tori is obtained.

Abstract

We solved the Schr{\"o}dinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus T^n = R^n / {\Lambda} is defined as a quotient of the Euclidean space R^n by an arbitrary n-dimensional lattice {\Lambda}. The lattice is not necessary either cubic or rectangular. The magnetic field is also arbitrary. However, we restrict ourselves within potential-free problems; the Schr{\"o}dinger operator is assumed to be the Laplace operator defined with the covariant derivative. We defined an algebra that characterizes the symmetry of the Laplacian and named it the magnetic algebra. We proved that the space of functions on which the Laplacian acts is an irreducible representation space of the magnetic algebra. In this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus. We developed a new method for Fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the Laplacian. All the eigenfunctions are given in explicit forms.