Delocalization and wave-packet dynamics in one-dimensional diluted Anderson models

We study the nature of one-electron eigen-states in a one-dimensional diluted Anderson model where every Anderson impurity is diluted by a periodic function $f(l)$ . Using renormalization group and transfer matrix techniques, we provide accurate estimates of the extended states which appear in this model, whose number depends on the symmetry of the diluting function $f(l)$. The density of states (DOS) for this model is also numerically obtained and its main features are related to the symmetries of the diluting function $f(l)$. Further, we show that the emergence of extended states promotes a sub-diffusive spread of an initially localized wave-packet.Comment: 6 pages, 6 figures, to appear in EPJ

On the rate of black hole binary mergers in galactic nuclei due to dynamical hardening

We assess the contribution of dynamical hardening by direct three-body scattering interactions to the rate of stellar-mass black hole binary (BHB) mergers in galactic nuclei. We derive an analytic model for the single-binary encounter rate in a nucleus with spherical and disk components hosting a super-massive black hole (SMBH). We determine the total number of encounters $N_{\rm GW}$ needed to harden a BHB to the point that inspiral due to gravitational wave emission occurs before the next three-body scattering event. This is done independently for both the spherical and disk components. Using a Monte Carlo approach, we refine our calculations for $N_{\rm GW}$ to include gravitational wave emission between scattering events. For astrophysically plausible models we find that typically $N_{\rm GW} \lesssim$ 10. We find two separate regimes for the efficient dynamical hardening of BHBs: (1) spherical star clusters with high central densities, low velocity dispersions and no significant Keplerian component; and (2) migration traps in disks around SMBHs lacking any significant spherical stellar component in the vicinity of the migration trap, which is expected due to effective orbital inclination reduction of any spherical population by the disk. We also find a weak correlation between the ratio of the second-order velocity moment to velocity dispersion in galactic nuclei and the rate of BHB mergers, where this ratio is a proxy for the ratio between the rotation- and dispersion-supported components. Because disks enforce planar interactions that are efficient in hardening BHBs, particularly in migration traps, they have high merger rates that can contribute significantly to the rate of BHB mergers detected by the advanced Laser Interferometer Gravitational-Wave Observatory.Comment: 13 pages, 9 figures, accepted for publication in MNRA