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 \hbox{$Q=-i\hbar \mathcal{S}^{\dagger}\frac{\ partial \mathcal{S}}{\partial E}$} Q = - i Latin small letter h with stroke † ∂ ∂E at the spectrum edge with a scattering matrix \hbox{$\mathcal{S}$} belonging to circular ensembles. The three classes of symmetry (β = 1,2 and 4) are 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 is given to the Wigner's time τ _w = Tr(Q) and its distribution is given analytically in the case of two-mode scattering. The results are presented exactly at all the Fermi energies without approximation and are tested numerically with an excellent precision.