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In a unit-spacing periodic spinless chain, self-paired momenta are $k=0$ and, when available, $k=\pi$. An anomalous pair of the same creation operator vanishes by the canonical anticommutation relations. Such a mode is a single normal number level, filled or empty according to its energy sign, rather than a two-distinct-mode Bogoliubov transformation. This excludes the paired-angle construction at those special momenta.

For a real squeezing parameter $\gamma$, the single-mode squeeze operator isIts exponent is skew-adjoint, so $S(\gamma )$ is a unitary operator, and it implements the Bogoliubov transformation

Without a preferred timelike Killing flow there is no canonical positive-frequency splitting. Different choices are related by a Bogoliubov transformation and generally define different particle notions and vacuum states.

A dilute condensed Bose system whose interaction is sufficiently weak for a small-depletion expansion. Its low-energy excitations are described by a Bogoliubov transformation rather than free particles. The homogeneous quadratic theory is controlled only while noncondensate occupation remains small compared with the condensate and its excitation energies are stable.