Diagonalization of bosonic quadratic Hamiltonians by Bogoliubov transformations

TL;DR

This paper establishes general and optimal conditions under which bosonic quadratic Hamiltonians, even with unbounded operators and infinite degrees of freedom, can be diagonalized using Bogoliubov transformations.

Contribution

It provides the first comprehensive set of criteria for diagonalizing bosonic quadratic Hamiltonians in complex quantum systems with unbounded operators.

Findings

Conditions for diagonalization are established and proven.

Results are optimal when one-body operators commute.

Applicable to systems with infinite degrees of freedom.

Abstract

We provide general conditions for which bosonic quadratic Hamiltonians on Fock spaces can be diagonalized by Bogoliubov transformations. Our results cover the case when quantum systems have infinite degrees of freedom and the associated one-body kinetic and paring operators are unbounded. Our sufficient conditions are optimal in the sense that they become necessary when the relevant one-body operators commute.