Laurent Bruneau, Jan Derezinski
Bogoliubov Hamiltonians and one parameter groups of Bogoliubov transformations
(375K, pdf)

ABSTRACT.  On the bosonic Fock space, a family of Bogoliubov transformations 
corresponding to a strongly continuous one-parameter group of symplectic 
maps $R(t)$ is considered. Under suitable assumptions on the generator $A$ of this group, which guarantee that the induced representations of CCR are unitarily equivalent for all time $t$, it is known that the unitary operator $U_{nat}(t)$ which implement this transformation gives a projective unitary representation of $R(t)$. Under rather general assumptions on the generator $A$, we prove that the corresponding Bogoliubov transformations can be implemented by a one-parameter 
group $U(t)$ of unitary operators. The generator of $U(t)$ will be called a Bogoliubov Hamiltonian. We will introduce two kinds of Bogoliubov 
Hamiltonians (type I and II) and give conditions so that they are well defined.