The bracketing theorem in the partitioning technique for solving the Schrodinger equation may be used in principle to determine upper and lower bounds to energy eigenvalues. Practical lower bounds of any accuracy desired may be evaluated by utilizing the properties of "inner projections" on finite manifolds in the Hilbert space. The method is here applied to the ground state and excited states of a Hamiltonian having a positive definite perturbation Even if inspiration is derived from the method of intermediate Hamiltonians, the final results are of bracketing type and independent of this approach. The method is numerically illustrated in some companying papers.
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