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

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 ($β=1$, 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 $τ_w=tr(Q)$ 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.

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