Localization and ergodicity in nonperturbative regions of energy eigenfunctions: from nearly integrable to chaotic

For a schematic shell model, which is conservative and possesses a chaotic classical limit, it is shown that the division of energy eigenfunctions into perturbative and nonperturbative parts is useful in the study of proper-ties such as localization and "ergodicity" of eigenfunctions. It is suggested that the properties should be studied in the nonperturbative regions and their neighborhood of the eigenfunctions. It is found that the average relative localization length of nonperturbative parts of eigenfunctions can characterize the behavior of the quantum system, in the process of the underlying classical system changing from mixed to chaotic. For nonperturbative parts of eigenfunctions whose relative localization length are close to one, good agreement is found between the distribution of their components and the prediction of random matrix theory. (C) 2001 Elsevier Science B.V. All rights reserved.