Self-consistent theory of lower bounds for eigenvalues

A rigorous practically applicable theory is presented for obtaining lower bounds to eigenvalues of Hermitian operators, whether the ground state or excited states. Algorithms are presented for computing "residual energies" whose magnitude is essential for the computation of the eigenvalues. Their practical application is possible due to the usage of the Lanczos method for creating a tridiagonal representation of the operator under study. The theory is self-consistent, in the sense that a lower bound for one state may be used to improve the lower bounds for others, and this is then used self-consistently until convergence. The theory is exemplified for a toy model of a quartic oscillator, where with only five states the relative error in the lower bound for the ground state is reduced to 6 · 10^-6, which is the same as the relative error of the least upper bound obtained with the same basis functions. The lower bound method presented in this paper suggests that lower bounds may become a staple of eigenvalue computations.