Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues

TL;DR

This paper establishes sharp bounds relating boundary norms of trial solutions to eigenvalue proximity and eigenfunction errors for the Dirichlet Laplacian, improving previous bounds and enabling high-frequency eigenvalue computations.

Contribution

It introduces boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues, with explicit constants and applications to high-frequency eigenvalue problems.

Findings

Bounds are sharp up to constants for all E above a small threshold.

Improves Moler--Payne bounds by a factor of  .

Numerical example achieves 14-digit accuracy for a high eigenvalue.

Abstract

We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain $\Omega\subset\RR^n$ with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E > 0$ and the spectrum $\{E_j \}$ in terms of the boundary $L^2$-norm of a normalized trial solution $u$ of the Helmholtz equation $(\Delta + E)u = 0$. We also bound the $L^2$-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all $E$ greater than a small constant, and improve upon the best-known bounds of Moler--Payne by a factor of the wavenumber $\sqrt{E}$. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes, of interest in its own right.