LOCALIZATION AND FLUCTUATIONS OF LOCAL SPECTRAL DENSITY ON TREE-LIKE STRUCTURES WITH LARGE CONNECTIVITY - APPLICATION TO THE QUASI-PARTICLELINE-SHAPE IN QUANTUM DOTS

We study fluctuations of the local density of states (LDOS) on a treel ike lattice with large branching number m. The average form of the loc al spectral function (at a given value of the random potential in the observation point) shows a crossover from the Lorentzian to a semicirc ular form at alpha similar to 1/m, where alpha=(V/W)(2), V is the typi cal value of the hopping matrix element, and W is the width of the dis tribution of random site energies. For alpha>1/m(2) the LDOS fluctuati ons (with respect to this average form) are weak. In the opposite case alpha<1/m(2), the fluctuations become strong and the average LDOS cea ses to be representative, which is related to the existence of the And erson transition at alpha(c) similar to 1/m(2)log(2)m. On the localize d side of the transition the spectrum is discrete and the LDOS is give n by a set of delta-like peaks. The effective number of components in this regime is given by 1/P, with P being the inverse participation ra tio. It is shown that P has in the transition point a limiting value P -c close to unity, 1-P-c similar to 1/logm, so that the system undergo es a transition directly from the deeply localized phase to the extend ed phase. On the side of delocalized states, the peaks in the LDOS bec ome broadened, with a width similar to exp{-const logm[(alpha-alpha(c) )/alpha(c)](-1/2)} being exponentially small near the transition point . We discuss the application of our results to the problem of the quas iparticle line shape in a finite Fermi system, as suggested recently b y Altshuler, Gefen, Kamenev, and Levitov [Phys. Rev. Lett. 78, 2803 (1 997)].