Derezinski J. Van Hove Hamiltonians---exactly solvable models of the infrared and ultraviolet problem---revised version. (68K, LATex 2e) ABSTRACT. Quadratic bosonic Hamiltonians with a linear perturbation are studied. Depending on the infrared and ultraviolet behavior of the perturbation, their properties are described from the point of view of spectral and scattering theory.