Quasi-orthogonality on the boundary for Euclidean Laplace eigenfunctions

Consider the Laplacian in a bounded domain in Rd 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(E1/2) centered about E, as E → ∞. 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.