Effective Hamiltonian and unitarity of the S matrix

Abstract

The properties of open quantum systems are described well by an effective Hamiltonian $\mathcal{H}$ that consists of two parts: the Hamiltonian $H$ of the closed system with discrete eigenstates and the coupling matrix $W$ between discrete states and continuum. The eigenvalues of $\mathcal{H}$ determine the poles of the $S$ matrix. The coupling matrix elements ${\tilde{W}}_k^{c{c}^{\prime }}$ between the eigenstates $k$ of $\mathcal{H}$ and the continuum may be very different from the coupling matrix elements $W_k^{c{c}^{\prime }}$ between the eigenstates of $H$ and the continuum. Due to the unitarity of the $S$ matrix, the $\TW_k^{cc'}$ depend on energy in a non-trivial manner, that conflicts with the assumptions of some approaches to reactions in the overlapping regime. Explicit expressions for the wave functions of the resonance states and for their phases in the neighbourhood of, respectively, avoided level crossings in the complex plane and double poles of the $S$ matrix are given.

17 pages, 7 figures

References in corpus (5)

• Observation of a Chiral State in a Microwave Cavity
• Random Matrices close to Hermitian or unitary: overview of methods and results
• On the statistics of resonances and non-orthogonal eigenfunctions in a model for single-channel chaotic scattering
• Is the concept of the non-Hermitian effective Hamiltonian relevant in the case of potential scattering?
• Exceptional points and double poles of the S matrix

Cited by in corpus (13)

• A review of progress in the physics of open quantum systems: theory and experiment
• Resonances in open quantum systems
• Nearby states in non-Hermitian quantum systems
• Open quantum systems and Dicke superradiance
• Phase rigidity and avoided level crossings in the complex energy plane
• Fano resonances in the overlapping regime
• Transmission zeros and ultrasensitive detection in complex systems
• Correlated behavior of conductance and phase rigidity in the transition from the weak-coupling to the strong-coupling regime
• Exceptional points in the scattering continuum
• Gain and loss in open quantum systems
• The brachistochrone problem in open quantum systems
• Open quantum systems with loss and gain
• Critical points in two-channel quantum systems