  Quadratic Hamiltonians are important in quantum field theory and quantum
statistical mechanics. Their general studies, which go back to the sixties, are
relatively incomplete for the fermionic case studied here. Following Berezin,
they are quadratic in the fermionic field and in this way well-defined
self-adjoint operators acting on the fermionic Fock space. We analyze their
diagonalization by applying a novel elliptic operator-valued differential
equations studied in a companion paper. This allows for their ($\mathrm{N}$-)
diagonalization under much weaker assumptions than before. Last but not least,
in 1994 Bach, Lieb and Solovej defined them to be generators of strongly
continuous unitary groups of Bogoliubov transformations. This is shown to be an
equivalent definition, as soon as the vacuum state belongs to the domain of
definition of these Hamiltonians. This second outcome is demonstrated to be
reminiscent to the celebrated Shale-Stinespring condition on Bogoliubov
transformations.
