# 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…

## 21 Citations

### Hamiltonian and linear-space structure for damped oscillators: II. Critical points

- Physics
- 2004

The eigenvector expansion developed in the preceding paper for a system of damped linear oscillators is extended to critical points, where eigenvectors merge and the time-evolution operator H assumes…

### SUSY transformations for quasinormal modes of open systems

- Physics
- 2001

Supersymmetry (SUSY) in quantum mechanics is extended from square-integrable states to those satisfying the outgoing-wave boundary condition, in a Klein–Gordon formulation. This boundary condition…

### SUSY Transformations for Quasinormal and Total-Transmission Modes of Open Systems

- Physics
- 1999

Quasinormal modes are the counterparts in open systems of normal modes in conservative systems; defined by outgoing-wave boundary conditions, they have complex eigenvalues. The conditions are studied…

### Hamiltonian and linear-space structure for damped oscillators: I. General theory

- Mathematics
- 2004

The phase space of N damped linear oscillators is endowed with a bilinear map under which the evolution operator is symmetric. This analogue of self-adjointness allows properties familiar from…

### Sensitivity of parameter estimation near the exceptional point of a non-Hermitian system

- PhysicsNew Journal of Physics
- 2019

The exceptional points (EPs) of non-Hermitian systems, where n different energy eigenstates merge into an identical one, have many intriguing properties that have no counterparts in Hermitian…

### Pareto optimal structures producing resonances of minimal decay under L1-type constraints

- Mathematics
- 2014

### Nonlinear Bang–Bang Eigenproblems and Optimization of Resonances in Layered Cavities

- Mathematics
- 2015

We study optimization of quasi-normal-eigenvalues $$\omega $$ω associated with the equation $$y^{\prime \prime } = -\omega ^2 B y $$y″=-ω2By of two-side open optical and mechanical resonators. The…

### An approach to the quantization of black-hole quasi-normal modes

- Physics
- 2013

In this work we derive the asymptotic quasi-normal modes of a BTZ black hole using a quantum field theoretic Lagrangian. The BTZ black hole is a very popular system in the context of…

### Lindbladians with multiple steady states: theory and applications

- Physics
- 2017

Markovian master equations, often called Liouvillians or Lindbladians, are used to describe decay and decoherence of a quantum system induced by that system's environment. While a natural environment…

### Linking quasi-normal and natural modes of an open cavity

- Physics
- 2010

The present paper proposes a comparison between the extinction theorem and the Sturm–Liouville theory approaches for calculating the electromagnetic (e.m.) field inside an optical cavity. We discuss…

## References

SHOWING 1-10 OF 73 REFERENCES

### Two-component eigenfunction expansion for open systems described by the wave equation II: linear space structure

- Physics
- 1997

For a broad class of open systems described by the wave equation, the eigenfunctions (which are quasinormal modes) provide a complete basis for simultaneously expanding outgoing wavefunctions . In…

### Quasinormal-mode expansion for waves in open systems

- Physics
- 1998

An open system is not conservative because energy can escape to the outside. As a result, the time-evolution operator is not Hermitian in the usual sense and the eigenfunctions (factorized solutions…

### TWO-COMPONENT EIGENFUNCTION EXPANSION FOR OPEN SYSTEMS DESCRIBED BY THE WAVE EQUATION. I: COMPLETENESS OF EXPANSION

- Environmental Science
- 1997

The concept of eigenfunction expansions for the wave equation is generalized to open systems, in which waves escape to the outside. These non-conservative systems are non-Hermitian in the usual…

### Second-quantization of open systems using quasinormal modes

- Physics
- 1998

The second-quantization of a scalar field in an open cavity is formulated, from first principles, in terms of the quasinormal modes (QNMs), which are the eigensolutions of the evolution equation that…

### Exactly solvable path integral for open cavities in terms of quasinormal modes

- Physics
- 2000

We evaluate the finite-temperature Euclidean phase-space path integral for the generating functional of a scalar field inside a leaky cavity. Provided the source is confined to the cavity, one can…

### Quasinormal mode expansion for linearized waves in gravitational systems.

- PhysicsPhysical review letters
- 1995

The quasinormal modes (QNM's) of gravitational systems modeled by the Klein-Gordon equation with effective potentials are studied in analogy to the QNM's of optical cavities, answering a conjecture by Price and Husain.

### Results on Certain Non‐Hermitian Hamiltonians

- Mathematics
- 1967

We present a few results on the spectral properties of a class of physically reasonable non‐Hermitian Hamiltonians. These theorems relate the spectral properties of a non‐self‐adjoint operator (of…

### Wave propagation in gravitational systems: Completeness of quasinormal modes.

- MathematicsPhysical review. D, Particles and fields
- 1996

This study opens up the possibility of using QNM expansions to analyse the behavior of waves in relativistic systems, even for systems whose QNM’s do not form a complete set, by introducing an infinitesimal change in the effective potential.