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We study the orthogonal polynomials and the Hankel determinants associatedwith Gaussian weight with two jump discontinuities. When the degree $n$ isfinite, the orthogonal polynomials and the Hankel determinants are shown to beconnected to the coupled Painlev\'e IV system. In the double scaling limit asthe jump discontinuities tend to the edge of the spectrum and the degree $n$grows to infinity, we establish the asymptotic expansions for the Hankeldeterminants and the orthogonal polynomials, which are expressed in terms ofsolutions of the coupled Painlev\'{e} II system. As applications, we re-derivethe recently found Tracy-Widom type expressions for the gap probability ofthere being no eigenvalues in a finite interval near the the extreme eigenvalueof large Gaussian unitary ensemble and the limiting conditional distribution ofthe largest eigenvalue in Gaussian unitary ensemble by considering a thinnedprocess.

Gaussian unitary ensemble with jump discontinuities and the coupled Painlev\'e II and IV systems