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

TL;DR

This paper explores how open wave systems can realize bi-orthogonal bases in dissipative systems, addressing Jordan block and orthonormalization issues through a generalized duality transformation involving higher-order poles.

Contribution

It introduces a generalized duality transformation with extra basis vectors to extend bi-orthogonal formalism in dissipative open wave systems.

Findings

Bi-orthogonal systems in open waves can face Jordan block and orthonormalization problems.

Higher-order poles in Green's functions are related to these issues.

A new transformation involving polynomials in time resolves these problems.

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.