A generalized expansion method for computing Laplace-Beltrami eigenfunctions on manifolds

TL;DR

This paper introduces a new numerical method for computing Laplace-Beltrami eigenfunctions on manifolds, leveraging a relaxation to the Schrödinger operator and basis projection, with proven spectral accuracy and applications in quantum billiards.

Contribution

It presents a novel generalized expansion method that ensures spectral exactness for eigenfunction computation on manifolds, advancing numerical techniques in geometric analysis.

Findings

Method achieves spectral exactness in eigenvalue computation.

Successfully applied to quantum billiards on manifolds.

Provides accurate eigenfunctions for complex geometries.

Abstract

Eigendecomposition of the Laplace-Beltrami operator is instrumental for a variety of applications from physics to data science. We develop a numerical method of computation of the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a smooth bounded domain based on the relaxation to the Schr\"odinger operator with finite potential on a Riemannian manifold and projection in a special basis. We prove spectral exactness of the method and provide examples of calculated results and applications, particularly, in quantum billiards on manifolds.