Lanczos method of tridiagonalization, Jacobi matrices and physics

Abstract

A survey of methods to determine the collective properties of the spectrum of a large-scale many-body system is given. They have been recently described and applied to a wide variety of fields in the physical sciences, from atomic and nuclear physics to solid state theory and geophysics. These methods use the classical Lanczos algorithm of tridiagonalization as a function-theoretic method and do not require the diagonalization of any matrix. The procedures outlined in this paper are able to express not only the N-dimensional Jacobi Hamiltonian of a physical system in terms of the spectral distribution ω 1 (E) of an arbitrary starting vector 1.1> of the reduced Hilbert space, but also the total density of states and its asymptotic limit (i.e. N→∞) as well as other densities of states of the system, e.g. that of the so-called group orbital in chemisorption theory. Also, procedures to calculate the total density of states of a Jacobi Hamiltonian and the asymptotical limit in terms of its matrix elements are given.