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We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on abounded domain $\Omega\subset\RR^n$ with piecewise smooth boundary. We boundthe 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 theHelmholtz equation $(\Delta + E)u = 0$. We also bound the $L^2$-norm of theerror of this trial solution from an eigenfunction. Both of these results aresharp up to constants, hold for all $E$ greater than a small constant, andimprove upon the best-known bounds of Moler--Payne by a factor of thewavenumber $\sqrt{E}$. One application is to the solution of eigenvalueproblems at high frequency, via, for example, the method of particularsolutions. In the case of planar, strictly star-shaped domains we give aninclusion bound where the constant is also sharp. We give explicit constants inthe theorems, and show a numerical example where an eigenvalue around the2500th is computed to 14 digits of relative accuracy. The proof makes use of anew quasi-orthogonality property of the boundary normal derivatives of theeigenmodes, of interest in its own right.

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