Transformation brackets for generalized Bogolyubov–boson transformations

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Abstract
We consider the transformation bi=Σsj=1 (λijaj+μijaj) between two sets of boson operators, with λij and μij complex. Closed formulas are derived for the transformation brackets connecting base states of the two sets. As an application, the general quadratic Hamiltonian in one dimension is diagonalized and two examples, involving time‐dependent real transformation and time‐independent complex transformation, are worked out.