Perturbative and non-perturbative parts of eigenstates and local spectral density of states: the Wigner band random matrix model

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A generalization of Brillouin-Wigner perturbation theory is applied numerically to the Wigner Band Random Matrix model. The perturbation theory tells that a perturbed energy eigenstate can be divided into a perturbative part and a non-perturbative part with the perturbative part expressed as a perturbation expansion. Numerically it is found that such a division is important in understanding many properties of both eigenstates and the so-called local spectral density of states (LDOS). For the average shape of eigenstates, its central part is found to be composed of its non-perturbative part and a region of its perturbative part, which is close to the non-perturbative part. A relationship between the average shape of eigenstates and that of LDOS can be explained. Numerical results also show that the transition for the average shape of LDOS from the Breit-Wigner form to the semicircle form is related to a qualitative change in some properties of the perturbation expansion of the perturbative parts of eigenstates. The transition for the half-width of the LDOS from quadratic dependence to linear dependence on the perturbation strength is accompanied by a transition of a similar form for the average size of the non-perturbative parts of eigenstates. For both transitions, the same critical perturbation strength λb has been found to play important roles. When perturbation strength is larger than λb, the average shape of LDOS obeys an approximate scaling law.