  We study the orthogonal polynomials and the Hankel determinants associated
with Gaussian weight with two jump discontinuities. When the degree $n$ is
finite, the orthogonal polynomials and the Hankel determinants are shown to be
connected to the coupled Painlev\'e IV system. In the double scaling limit as
the jump discontinuities tend to the edge of the spectrum and the degree $n$
grows to infinity, we establish the asymptotic expansions for the Hankel
determinants and the orthogonal polynomials, which are expressed in terms of
solutions of the coupled Painlev\'{e} II system. As applications, we re-derive
the recently found Tracy-Widom type expressions for the gap probability of
there being no eigenvalues in a finite interval near the the extreme eigenvalue
of large Gaussian unitary ensemble and the limiting conditional distribution of
the largest eigenvalue in Gaussian unitary ensemble by considering a thinned
process.
