Bounds to the extreme eigenvalues of the Lanczos Hamiltonian of a quantum system
F. J. Gálvez, J. S. Dehesa · Journal of Physics A Mathematical and General · 1985
The Lanczos method is an iterative scheme which when applied to the Hamiltonian H of a quantum system and starting from an initial vector (which is, in principle, an arbitrary vector of the associated Hilbert space) produces very good approximations to the eigenvalues at the extreme ends of the spectrum after a few iterations. In this comment the authors derive analytically a lower bound to the smallest eigenvalue and an upper bound to the largest eigenvalue of the Lanczos Hamiltonian of the system in terms of the expectation of H evaluated at the initial vector.