Lower Bounds to Eigenvalues of the Schrödinger Equation. I. Formalism for Excited States

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Abstract
A procedure is presented for the calculation of lower bounds to the excited states of a Hamiltonian of the type H=H0+V, where H is bounded below and self-adjoint, V is positive definite, and H0 has a known set of eigenfunctions. This procedure is derived using the partitioning technique. A detailed study is made into the effect that replacing the reaction operator t(ε) by its inner projection onto some linear manifold f has on the curves of the multivalued bracketing function. From this analysis it is shown for the first time that when the linear manifold, onto which the projection of t(ε) is made, is of the type f=t(ε)−1/2 (ε−H0) j, the bracketing curves are no longer monotonically descending when plotted as a function of ε. This result is very important in the calculation of lower bounds to excited states.