Generalized inequalities for the Bogoliubov-Duhamel inner product with applications in the Approximating Hamiltonian Method

TL;DR

This paper develops generalized inequalities for the Bogoliubov-Duhamel inner product, providing bounds on fluctuations in many-particle systems, with applications demonstrated on exactly solvable models involving bosons and matter.

Contribution

It introduces a unified set of inequalities that extend existing bounds, applicable to a broad class of models within the Approximating Hamiltonian framework.

Findings

Derived inequalities generalize all known bounds in the context.

Upper bounds maintain the same form and order with respect to N.

Applications to models with boson interactions demonstrate the inequalities' effectiveness.

Abstract

Infinite sets of inequalities which generalize all the known inequalities that can be used in the majorization step of the Approximating Hamiltonian method are derived. They provide upper bounds on the difference between the quadratic fluctuations of intensive observables of a $N$-particle system and the corresponding Bogoliubov-Duhamel inner product. The novel feature is that, under sufficiently mild conditions, the upper bounds have the same form and order of magnitude with respect to $N$ for all the quantities derived by a finite number of commutations of an original intensive observable with the Hamiltonian. The results are illustrated on two types of exactly solvable model systems: one with bounded separable attraction and the other containing interaction of a boson field with matter.