  This paper studies the Hankel determinants generated by a discontinuous
Gaussian weight with one and two jumps. It is an extension of Chen and
Pruessner \cite{Chen2005}, in which they studied the discontinuous Gaussian
weight with a single jump. By using the ladder operator approach, we obtain a
series of difference and differential equations to describe the Hankel
determinant for the single jump case. These equations include the Chazy II
equation, continuous and discrete Painlev\'{e} IV. In addition, we consider the
large $n$ behavior of the corresponding orthogonal polynomials and prove that
they satisfy the biconfluent Heun equation. We also consider the jump at the
edge under a double scaling, from which a Painlev\'{e} XXXIV appeared.
Furthermore, we study the Gaussian weight with two jumps, and show that a
quantity related to the Hankel determinant satisfies a two variables'
generalization of the Jimbo-Miwa-Okamoto $\sigma$ form of the Painlev\'{e} IV.
