Nonperturbative and perturbative parts of energy eigenfunctions: a three-orbital schematic shell model

We study the division of components of energy eigenfunctions, as the expansion of perturbed states in unperturbed states, into nonperturbative and perturbative parts in a three-orbital schematic shell model possessing a chaotic classical limit, the Hamiltonian of which is composed of a Hamiltonian of noninteracting particles and a perturbation. The perturbative parts of eigenfunctions are expanded in a convergent perturbation expansion by making use of the nonperturbative parts. The division is shown to have the property that, when the underlying classical system is chaotic, the statistics of the components of the nonperturbative parts whose relative localization length are close to 1 is in agreement with the prediction of random-matrix theory. When the underlying classical system is mixed, main bodies of most of the eigenfunctions are found to occupy parts of their nonperturbative regions, with some of the rest of the eigenfunctions being "ergodic" in their nonperturbative regions due to avoided level crossings. In case of the classical system being chaotic, most of the eigenfunctions are found "ergodic" or almost "ergodic" in their nonperturbative regions. Numerical results show that the average relative localization length of nonperturbative parts of eigenfunctions is useful in characterizing the behavior of the quantum system in the process of the underlying classical system changing from a mixed system to a chaotic one.