Statistics of eigenfunctions in open chaotic systems: a perturbative approach

TL;DR

This paper analytically derives the probability distribution of the complexness parameter for resonance eigenstates in open chaotic systems, using random matrix theory, and explores effects of spectral fluctuations.

Contribution

It introduces a perturbative approach to analytically compute the distribution of the complexness parameter in open chaotic systems, considering both rigid and fluctuating spectra.

Findings

Derived the probability distribution for the complexness parameter.

Analyzed the impact of spectral fluctuations on the distribution.

Provided analytical results for systems with and without spectral rigidity.

Abstract

We investigate the statistical properties of the complexness parameter which characterizes uniquely complexness (biorthogonality) of resonance eigenstates of open chaotic systems. Specifying to the regime of isolated resonances, we apply the random matrix theory to the effective Hamiltonian formalism and derive analytically the probability distribution of the complexness parameter for two statistical ensembles describing the systems invariant under time reversal. For those with rigid spectra, we consider a Hamiltonian characterized by a picket-fence spectrum without spectral fluctuations. Then, in the more realistic case of a Hamiltonian described by the Gaussian Orthogonal Ensemble, we reveal and discuss the r\^ole of spectral fluctuations.