Quantum systems describing the interaction between particles and fields are generally complex, and their analysis is often difficult. Among such systems, models with quadratic interactions are believed to be diagonalizable by Bogoliubov transformations, and many works have been devoted to their study.

In this talk, we present a method for algebraically diagonalizing abstract pair models by means of bosonic Bogoliubov transformations. The resulting Hamiltonians are represented as the second quantization of explicit one-particle Hamiltonians, and an explicit formula for the ground state energy is obtained.

As examples, we discuss a harmonic oscillator coupled to a quantum Bose field and the Pauli–Fierz model in the dipole approximation. Through this method, we can proceed to the spectral analysis of the models. In particular, we establish analyticity with respect to the parameters, leading to a rigorous justification of the perturbation expansion.