05-398 Laurent Bruneau, Jan Derezinski

Bogoliubov Hamiltonians and one parameter groups of Bogoliubov transformations (375K, pdf) Nov 22, 05

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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.

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