Spectrum and normal modes of non-hermitian quadratic boson operators

We analyze the spectrum and normal mode representation of general quadratic bosonic forms $H$ not necessarily hermitian. It is shown that in the one-dimensional case such forms exhibit either an harmonic regime where both $H$ and $H^\dagger$ have a discrete spectrum with biorthogonal eigenstates, and a coherent-like regime where either $H$ or $H^\dagger$ 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 to $N$-dimensional quadratic systems is as well discussed.