In the present paper a method by Lehmann-Maehly and Goerisch is extended to self-adjoint eigenvalue problems with arbitrary essential spectrum. This extension is obtained by consequently making use of the local character of the method. In this way, upper and lower bounds to all isolated eigenvalues are derived. In our proofs, the close relationship to Wielandt’s inverse iteration becomes quite obvious.
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S. Zimmermann, U. Mertins, Variational Bounds to Eigenvalues of Self-Adjoint Eigenvalue Problems with Arbitrary Spectrum. Z. Anal. Anwend. 14 (1995), no. 2, pp. 327–345