1-level correlation functions of disturbed unitary ensembles

In the random matrix theory, the 1-level correlation functions Rn??1(x) (?? is generally called Dyson's index) can be explained as the distribution density of energy levels which can be found nearby x. As yet, its limit behavior is still remarkably noticed by many mathematicians and physicists. In the case of unitary ensemble (i.e. ?? = 2), the 1-level correlation functions Rn2(x) is closely related to the weight function ??(x) in the classical orthogonal polynomial theory. For the simplication of statement, only the Gaussian unitary ensemble case is considered. The authors find that in weak sense, the limit behavior of Rn21(x) is not disturbed by appending a 'good' nontrivial multiplicative factor to ??(x).EI05671-6784