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Dissipative systems can be described in terms of non-hermitian hamiltoniansH, whose left eigenvectors f^j and right eigenvectors f_j form a bi-orthogonalsystem. Bi-orthogonal systems could suffer from two difficulties. (a) If theeigenvectors do not span the whole space, then H can only be diagonalized toblocks (the Jordan-block problem). (b) Normalization would not be possible andmany familiar-looking formulas would fail if (f^j,f_j) = 0 for some j (theorthonormalization problem). Waves in open systems provide a well-foundedrealization of a bi-orthogonal system, and it is shown that these two problemscan indeed occur and are both related to higher-order poles in thefrequency-domain Green's function. The resolution is then given by introducinga generalized duality transformation involving extra basis vectors, whose timeevolution is modified by polynomials in the time t. One thus obtains anontrivial extension of the bi-orthogonal formalism for dissipative systems.Comment: REVTeX, 25 pages, submitted to Phys. Rev.

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