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On the quality of complementary bounds for eigenvalues

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Abstract

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.

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Acknowledgments

Financial support was provided by the Engineering and Physical Sciences Research Council (grant number EP/I00761X/1) and King Abdulaziz University. We kindly thank the referee for so many useful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Lyonell Boulton

  2. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box. 80203, Jeddah, 21589, Saudi Arabia

    Aatef Hobiny

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  1. Lyonell Boulton
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  2. Aatef Hobiny
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Correspondence to Lyonell Boulton.

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Boulton, L., Hobiny, A. On the quality of complementary bounds for eigenvalues. Calcolo 52, 577–601 (2015). https://doi.org/10.1007/s10092-014-0131-y

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  • DOI: https://doi.org/10.1007/s10092-014-0131-y

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