The Lanczos algorithm for extensive many-body systems in the thermodynamiclimit

We establish rigorously the scaling properties of the Lanczos process appli ed to an arbitrary extensive many-body system which is carried to convergen ce n --> infinity and the thermodynamic limit N --> infinity taken. In this limit the solution for the limiting Lanczos coefficients are found exactly and generally through two equivalent sets of equations, given initial know ledge of the exact cumulant generating function. The measure and the orthog onal polynomial system associated with the Lanczos process in this regime a re also given explicitly. Some important representations of these Lanczos f unctions are given, including Taylor series expansions, and theorems contro ling their general properties are proven. (C) 1999 American Institute of Ph ysics. [S0022-2488(99)01510-8].