Spectrum and normal modes of non-Hermitian quadratic boson operators

Abstract

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 a harmonic regime where both H and H† have a discrete spectrum with biorthogonal eigenstates, and a coherent-like regime where either H or H† have a continuous complex twofold degenerate spectrum, while its adjoint has no convergent eigenstates. These regimes reflect the nature of the pertinent normal boson operators. Nondiagonalizable cases as well critical boundary sectors separating these regimes are also analyzed. The extension to N-dimensional quadratic systems is also discussed.