Simple rule for complex quantum systems

Abstract

The exact rule describing the transformation of spectra of multilevel quantum systems perturbed by a random matrix has been found with the help of the all-order perturbation theory. For systems with a large number of levels this rule has a simple form, resembling the second-order perturbation theory. It immediately gives the well-known semicircle Wigner distribution of the density of states and the double-hump distribution for binary alloys. The spectra of small atomic or molecular clusters resulting from the perturbation of a small number of identical levels are also given among some other examples. Discontinuities of the derivatives of the energy spectra and a nonexponential time evolution are shown to be the typical features of large multilevel systems perturbed by a random matrix.