Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems

@article{Brink1999JordanBA,
  title={Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems},
  author={Alec Maassen van den Brink and Kenneth R. Young},
  journal={Journal of Physics A},
  year={1999},
  volume={34},
  pages={2607-2624}
}
Dissipation can sometimes be described by a non-Hermitian Hamiltonian H, whose left and right eigenvectors {f j, fj} form a bi-orthogonal basis (BB). For waves in a class of open systems, this is known to lead to exact, complete BB expansions if f j| fj≠0 for all j. If not, normalization seems impossible and many familiar formulae fail; examples are given. The problem is related to the merging of eigenmodes, so that H can only be diagonalized to Jordan blocks. The resolution involves a… 

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