Abstract

Relative Lehman Boounds for Restricted Eigenvalue Problems C. Beattie Inequalities are presented for the inertia of a quadratic form restricted to arbitrary subspaces contained within its domain of definition. These inequalities lead to novel variational characterizations of eigenvalues of self-adjoint, semi-bounded operators, such as are commonly associated with boundary value problems in engineering and mathematical physics, and are the basis of a new approach for computing rigorous lower bounds to these eigenvalues. The necessary {\it a priori}spectral information is more accessible than for other methods and in the case the restricting subspace has a trivial complement this approach reduces to Lehmann's method. An example utilizing Birman-Schwinger bounds to extract base problem information for molecular hydrogen is described. The bounds obtained are complementary to those obtainable by the ubiquitous Rayleigh-Ritz procedure and together provide eigenvalue estimates with absolute error bounds.

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