On the quality of complementary bounds for eigenvalues

A concrete formulation of the Lehmann–Maehly–Goerisch method for semi-definite self-adjoint operators with compact resolvent is considered. Precise rates of convergence are determined in terms of how well the trial spaces capture the spectral subspace of the operator. Optimality of the choice of a shift parameter which is intrinsic to the method is also examined. The main theoretical findings are illustrated by means of a few numerical experiments involving one-dimensional Schrödinger operators.