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Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems
Mathematical Physics · Submitted 25 May 1999
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim One thus obtains a nontrivial extension of the bi-orthogonal formalism for dissipative systems.
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The abstract, sentence by sentence
Dissipative systems can be described in terms of non-hermitian hamiltonians H, whose left eigenvectors f^j and right eigenvectors f_j form a bi-orthogonal system. Bi-orthogonal systems could suffer from two difficulties. (a) If the eigenvectors do not span the whole space, then H can only be diagonalized to blocks (the Jordan-block problem). (b) Normalization would not be possible and many familiar-looking formulas would fail if (f^j,f_j) = 0 for some j (the orthonormalization problem). Waves in open systems provide a well-founded realization of a bi-orthogonal system, and it is shown that these two problems can indeed occur and are both related to higher-order poles in the frequency-domain Green's function. The resolution is then given by introducing a generalized duality transformation involving extra basis vectors, whose time evolution is modified by polynomials in the time t. One thus obtains a nontrivial extension of the bi-orthogonal formalism for dissipative systems.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Dissipative systems can be described in terms of non-hermitian hamiltonians H, whose left eigenvectors f^j and right eigenvectors f_j form a bi-orthogonal system. | background 1.00 | 0.06 | 0.03 | asserted 0.98 |
| s1 | Bi-orthogonal systems could suffer from two difficulties. | background 0.97 | 0.07 | 0.02 | hedged possibility 1.00 |
| s2 | (a) If the eigenvectors do not span the whole space, then H can only be diagonalized to blocks (the Jordan-block problem). | background 0.98 | 0.10 | 0.04 | asserted 0.60 |
| s3 | (b) Normalization would not be possible and many familiar-looking formulas would fail if (f^j,f_j) = 0 for some j (the orthonormalization problem). | background 0.98 | 0.11 | 0.04 | hedged possibility 0.80 |
| s4 | Waves in open systems provide a well-founded realization of a bi-orthogonal system, and it is shown that these two problems can indeed occur and are both related to higher-order poles in the frequency-domain Green's function. | result 0.63 | 0.87 | 0.55 | asserted 1.00 |
| s5 | The resolution is then given by introducing a generalized duality transformation involving extra basis vectors, whose time evolution is modified by polynomials in the time t. | method 0.62 | 0.88 | 0.41 | asserted 1.00 |
| s6 | One thus obtains a nontrivial extension of the bi-orthogonal formalism for dissipative systems. | interpretation 0.54 | 0.86 | 0.50 | asserted 0.99 |
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Original abstract
Dissipative systems can be described in terms of non-hermitian hamiltonians H, whose left eigenvectors f^j and right eigenvectors f_j form a bi-orthogonal system. Bi-orthogonal systems could suffer from two difficulties. (a) If the eigenvectors do not span the whole space, then H can only be diagonalized to blocks (the Jordan-block problem). (b) Normalization would not be possible and many familiar-looking formulas would fail if (f^j,f_j) = 0 for some j (the orthonormalization problem). Waves in open systems provide a well-founded realization of a bi-orthogonal system, and it is shown that these two problems can indeed occur and are both related to higher-order poles in the frequency-domain Green's function. The resolution is then given by introducing a generalized duality transformation involving extra basis vectors, whose time evolution is modified by polynomials in the time t. One thus obtains a nontrivial extension of the bi-orthogonal formalism for dissipative systems.
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