Time delay matrix at the spectrum edge and the minimal chaotic cavities

TL;DR

This paper derives the distribution of proper delay times at the spectrum edge for minimal chaotic cavities, analyzing symmetry classes and providing exact analytical results for the Wigner time distribution in two-mode scattering.

Contribution

It introduces a new universality class at the spectrum edge and provides exact analytical distributions for delay times, including the Wigner time, in minimal chaotic cavities.

Findings

Distribution of proper delay times at the spectrum edge derived.

Analytical expression for Wigner time distribution in two-mode scattering.

Numerical tests confirm the theoretical predictions with high precision.

Abstract

Using the concept of minimal chaotic cavities, we give the distribution of the proper delay times of $Q=-i\hbar S^\dagger \frac{\partial S}{\partial E}$ at the spectrum edge with a scattering matrix $S$ belonging to circular ensembles CE. The three classes of symmetry ($\beta=1$, 2 and 4) will be analyzed to show how it differs from the distribution obtained in the bulk of the spectrum. In this new class of universality at the spectrum edge, more attention will be given to the Wigner time $\tau_w=tr(Q)$ and its distribution will be given analytically in the case of 2 modes scattering. The results will be presented exactly at all the Fermi energies without any approximation. All this will be tested numerically with an excellent precision.