Improved Convergence Rates for Intermediate Problems

Improved convergence rate estimates are derived for a variant of Aronszajn-type intermediate problems that is both computationally feasible and convergent for problems with nontrivial essential spectra. In a previous paper the authors obtained rate of convergence estimates for this method in terms of containment gaps between subspaces. In the present work, techniques for estimating relatively unbounded perturbations are refined in order to apply the Kato-Temple inequalities. This yields convergence rates for the intermediate operator eigenvalues in terms of squares of containment gaps between subspaces. Convergence rate estimates are also obtained for the intermediate problem eigenvectors, and comparisons are made with previously known results for the method of special choice.