Diagonalization of the quadratic boson Hamiltonian with zero modes

As a coping-stone of our investigations concerning the general quadratic hamiltonian in a finite set of (boson or fermion) creation and annihilation operators we consider the homogeneous quadratic boson hamiltonian with zero modes. It is proved that zero (soft) modes, as they are connected with such hamiltonians, can only be of two types. The first type, to which we shall refer as a proper soft mode, corresponds to a spectrum with only one (zero-energy) level of infinite degeneracy. The second type, an improper (or spurious) soft mode, corresponds to a spectrum which is twofold degenerate and which consists of the continuum running from 0 to +∞. A simple method is given to determine the values of the (positive) mode energies and the number of both proper and improper soft modes from the coefficients in the hamiltonian. The algorithm presented in the preceding paper can be used for the construction of a homogeneous linear transformation of the boson operators which carries the hamiltonian into the standard (separated) form. Also the quadratic hamiltonian with a linear part is diagonalized, in particular the hamiltonian whose homogeneous quadratic part contains zero modes.