Boundary Quasi-Orthogonality and Sharp Inclusion Bounds for Large Dirichlet Eigenvalues

TLDR: In this paper, the distance between an arbitrary parameter and the spectrum of the Dirichlet Laplacian was shown to be at most a constant factor larger than the operator norm of any one individual term.

Abstract: We study eigenfunctions $\phi_j$ and eigenvalues $E_j$ of the Dirichlet Laplacian on a bounded domain $\Omega\subset\mathbb{R}^n$ 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 (Theorem 1.3), of interest in its own right. Namely, the operator norm of the sum of rank 1 operators $\partial_n\phi_j\langle\partial_n\phi_j,\cdot\rangle$ over all $E_j$ in a spectral window of width $\sqrt{E}$—a sum with about $E^{(n-1)/2}$ terms—is at most a constant factor (independent of $E$) larger than the operator norm of any one individual term.