The Lanczos algorithm for extensive many-body systems in the thermodynamic limit

Abstract

We establish rigorously the scaling properties of the Lanczos process applied to an arbitrary extensive many-body system which is carried to convergence n→∞ and the thermodynamic limit N→∞ taken. In this limit the solution for the limiting Lanczos coefficients are found exactly and generally through two equivalent sets of equations, given initial knowledge of the exact cumulant generating function. The measure and the orthogonal polynomial system associated with the Lanczos process in this regime are also given explicitly. Some important representations of these Lanczos functions are given, including Taylor series expansions, and theorems controling their general properties are proven.