Spectrum and normal modes of non-hermitian quadratic boson operators

Abstract

We analyze the spectrum and normal mode representation of general quadratic bosonic formsHnot necessarily hermitian. It is shown that in the one-dimensional case such forms exhibit either an harmonic regime where bothHandH^†have a discrete spectrum with biorthogonal eigenstates, and a coherent-like regime where eitherHorH†have a continuous complex two-fold degenerate spectrum, while its adjoint has no convergent eigenstates. These regimes reflect the nature of the pertinent normal boson operators. Non-diagonalizable cases as well critical boundary sectors separating these regimes are also analyzed. The extension toN-dimensional quadratic systems is as well discussed.