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

We study eigenfunctions φj and eigenvalues Ej of the Dirichlet Laplacian on a bounded domain Ω ⊂ Rn with piecewise smooth boundary. We bound the distance between an arbitrary parameter E > 0 and the spectrum {Ej} in terms of the boundary L2-norm of a normalized trial solution u of the Helmholtz equation (∆ + E)u = 0. We also bound the L2-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 √ 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 below), of interest in its own right. Namely, the operator norm of the sum of rank 1 operators ∂nφj〈∂nφj , ·〉 over all Ej in a spectral window of width √ 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.