The Lanczos Algorithm for extensive Many-Body Systems in the Thermodynamic Limit

TL;DR

This paper rigorously analyzes the scaling behavior of the Lanczos algorithm when applied to extensive many-body systems in the thermodynamic limit, providing exact solutions and explicit representations.

Contribution

It introduces a rigorous framework for understanding the Lanczos process in the thermodynamic limit, deriving exact limiting coefficients and associated measures.

Findings

Exact solutions for limiting Lanczos coefficients

Explicit measures and orthogonal polynomial systems

Theorems on properties of Lanczos functions

Abstract

We establish rigourously the scaling properties of the Lanczos process applied to an arbitrary extensive Many-Body System which is carried to convergence n to infinity and the thermodynamic limit N to 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 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 controlling their general properties are proven.