A Lower Bound Procedure for Energy Eigenvalues

Abstract

A lower bound procedure for energy eigenvalues based on the method of intermediate problems is given. A projection technique is used to construct a family of operators smaller than a given Hamiltonian whose eigenvalues are lower bounds to those of the given Hamiltonian. By a particular choice of subspaces associated with the projections it is possible to construct the family in such a way that certain members may have an eigenvalue coinciding with one of the real eigenvalues of a nonlinear but finite matrix eigenvalue problem. Application to the helium atom ground state indicates that the procedure may be more efficient than the procedures customarily used.