  In this paper, we study the Hankel determinant generated by a singularly
perturbed Gaussian weight $$
w(x,t)=\mathrm{e}^{-x^{2}-\frac{t}{x^{2}}},\;\;x\in(-\infty, \infty),\;\;t>0.
$$ By using the ladder operator approach associated with the orthogonal
polynomials, we show that the logarithmic derivative of the Hankel determinant
satisfies both a non-linear second order difference equation and a non-linear
second order differential equation. The Hankel determinant also admits an
integral representation involving a Painlev\'e III$'$. Furthermore, we consider
the asymptotics of the Hankel determinant under a double scaling, i.e.
$n\rightarrow\infty$ and $t\rightarrow 0$ such that $s=(2n+1)t$ is fixed. The
asymptotic expansions of the scaled Hankel determinant for large $s$ and small
$s$ are established, from which Dyson's constant appears.
