Under the condition that a certain hermitian operator has a self-adjoint extension a necessary and sufficient condition that a bilinear Fermion Hamiltonian can be diagonalized by a Bogoliubov transformation is obtained. Under the same assumption, any bilinear Fermion Hamiltonian can be diagonalized in a slightly extended sense by an extended Bogoliubov transformation. The meaning of this diagonalization from the view point of the Clifford C* algebra is discussed. It is shown that a parallel treatment is possible for a bilinear Boson Hamiltonian (with complications concerning unbounded operators and an indefinite metric) if a spectral theory of pseudo hermitian operator on a Hilbert space of an indefinite metric hold in parallel with that of definite metric Hilbert space.
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Huzihiro Araki, On the Diagonalization of a Bilinear Hamiltonian by a Bogoliubov Transformation. Publ. Res. Inst. Math. Sci. 4 (1968), no. 2, pp. 387–412DOI 10.2977/PRIMS/1195194882