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

Dissipation can sometimes be described by a non-Hermitian Hamiltonian H, wh ose left and right eigenvectors {f(j), f(j)} form a bi-orthogonal basis (BB ). For waves ina class of open systems, this is known to lead to exact, com plete BB expansions if [f(j) \f(j)] not equal 0 for all j. If not, normaliz ation 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 generalized BB co ntaining extra vectors, whose dynamics are modified by polynomials in the t ime t. The splitting of merged modes under a perturbation is also treated. One thus obtains a non-trivial extension of the BB formalism for open syste ms.