Delay-Time and Thermopower Distributions at the Spectrum Edges of a Chaotic Scatterer

We study chaotic scattering outside the wide band limit, as the Fermi energy $E_F$ approaches the band edges $E_B$ of a one-dimensional lattice embedding a scattering region of M sites. We show that the delay-time and thermopower distributions differ near the edges from the universal expressions valid in the bulk. To obtain the asymptotic universal forms of these edge distributions, one must keep constant the energy distance $E_F-E_B$ measured in units of the same energy scale proportional to $\propto M^{-1\mathrm{/}3}$ which is used for rescaling the energy level spacings at the spectrum edges of large Gaussian matrices. In particular the delay-time and the thermopower have the same universal edge distributions for arbitrary M as those for an M=2 scatterer, which we obtain analytically.