Extremal Properties of Resonance Eigenvalues

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Abstract
The minimax theorem for the eigenvalues of Hermitian operators is reviewed and its inverse (maxmini) theorem is derived. The connection between the Hylleraas-Undheim theorem (HUT) and the minimax theorem is clarified, and an extension of the HUT to compound resonance states is obtained. Thus, rigorous upper bounds on the resonance energies may be obtained without the explicit use of the exact open-channel projection operators. The exchange and rearrangement problems are also considered.