Quasi-orthogonality on the boundary for Euclidean Laplace eigenfunctions

TL;DR

This paper proves that Laplace eigenfunctions in bounded domains exhibit quasi-orthogonality on the boundary asymptotically, enabling more efficient numerical solutions for large eigenvalues without relying on domain ergodicity assumptions.

Contribution

It establishes boundary quasi-orthogonality of Laplace eigenfunctions for general boundary conditions, providing a rigorous foundation for a scaling numerical method.

Findings

Eigenfunctions are quasi-orthogonal on the boundary asymptotically.

The result holds independently of the domain's geodesic flow.

Boundary quasi-orthogonality improves numerical eigenproblem solutions.

Abstract

Consider the Laplacian in a bounded domain in R^d with general (mixed) homogeneous boundary conditions. We prove that its eigenfunctions are `quasi-orthogonal' on the boundary with respect to a certain norm. Boundary orthogonality is proved asymptotically within a narrow eigenvalue window of width o(E^{1/2}) centered about E, as E->infinity. For the special case of Dirichlet boundary conditions, the normal-derivative functions are quasi-orthogonal on the boundary with respect to the geometric weight function r.n. The result is independent of any quantum ergodicity assumptions and hence of the nature of the domain's geodesic flow; however if this is ergodic then heuristic semiclassical results suggest an improved asymptotic estimate. Boundary quasi-orthogonality is the key to a highly efficient `scaling method' for numerical solution of the Laplace eigenproblem at large eigenvalue. One of the main results of this paper is then to place this method on a more rigorous footing.